ontology
type: Ontology
version info: 2.5.0

kurtosis of Sinh-Arcsinh-2
type: mathematical object
has name: kurtosis of Sinh-Arcsinh-2
has code name: SinhArcsinh2.kurtosis
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short name: kurtosis
has short code name: kurtosis

shape parameter of Standard-Power-1
type: mathematical object
has name: shape parameter of Standard-Power-1
has code name: StandardPower1.shape
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape parameter
has short code name: shape

noncentrality parameter of Noncentral-chi-squared-1
type: mathematical object
has name: noncentrality parameter of Noncentral-chi-squared-1
has code name: NoncentralChiSquared1.noncentrality
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short name: noncentrality parameter
has short code name: noncentrality

variance of Inverse-Gamma 1
type: mathematical object
has name: variance of Inverse-Gamma 1
has Latex expression: \frac{\beta^2}{(\alpha-1)^2(\alpha-2)} \text{ for } \alpha > 2

mode of Logistic 1
type: mathematical object
has name: mode of Logistic 1
has Latex expression: \mu

mode of Exponential 1
type: mathematical object
has name: mode of Exponential 1
has Latex expression: 0

variance of Wigner Semicircle 1
type: mathematical object
has name: variance of Wigner Semicircle 1
has Latex expression: R^2/4

mean of GeneralizedPoisson2
type: mathematical object
has name: mean of GeneralizedPoisson2
has code name: GeneralizedPoisson2.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu > 0
has short name: mean
has short code name: mean

PDF of LKJ Correlation 2
type: probability density function
has name: PDF of LKJ Correlation 2
has Latex expression: \propto |J| \det(LL^T)^{(\eta-1)} = \prod_{k=2}^{K}L_{kk}^{K-k+2\eta-2}

variance of Generalized Poisson 3
type: mathematical object
has name: variance of Generalized Poisson 3
has Latex expression: \mu (1+\alpha \mu)^2

PDF of Multivariate Normal 1
type: probability density function
has name: PDF of Multivariate Normal 1
has Latex expression: (2\pi)^{-\frac{k}{2}}|\Sigma|^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(x-\mu)'\Sigma^{-1}(x-\mu) }

PDF of Inverse Gaussian 1
type: probability density function
has name: PDF of Inverse Gaussian 1
has Latex expression: \sqrt{\frac{\lambda}{2\pi x^3}} \exp\Big(-\frac{\lambda}{2\mu^2 x}(x-\mu)^2\Big)
has R code expression: sqrt(lambda/(2*pi*x^3) ) * exp(-lambda/(2*mu^2 x) * (x-mu)^2)

PDF of Chi 1
type: probability density function
has name: PDF of Chi 1
has Latex expression: \frac{2^{1-k/2} x^{k-1} e^{-x^2 / 2} }{\Gamma\left(\frac{k}{2}\right)}
has R code expression: 2^(1-k/2) * x^(k-1) * exp(-x^2/2) / gamma(k/2)

number of failures of Negative-Binomial-4
type: mathematical object
has name: number of failures of Negative-Binomial-4
has code name: NegativeBinomial4.numberOfFailures
has Latex symbol: r
parameter has definition expressed in plain text: r > 0, r \in N
has short name: number of failures
has short code name: numberOfFailures

variance of Trapezoidal 1
type: mathematical object
has name: variance of Trapezoidal 1
has Latex expression: \dfrac{2^2}{72(d+c-a-b)^2} \big(d^4+2cd^3-3bd^3-3ad^3 \\ -3bcd^2 - 3acd^2 + 4b^2d^2 + 4abd^2 \\ +4a^2d^2 + 2c^3d - 3bc^2d - 3ac^2d \\ +4b^2cd + 4abcd + 4a^2cd - 3b^3d - 3ab^2d \\ -3a^2bd - 3a^3d + c^4 - 3bc^3 - 3ac^3 + 4b^2c^2 \\ +4abc^2 + 4a^2c^2 - 3b^3c - 3ab^2c - 3a^2bc \\ -3a^3c + b^4 + 2ab^3 + 2a^3b + a4 \big)

PDF of Normal 3
type: probability density function
has name: PDF of Normal 3
has Latex expression: \sqrt{\frac{\tau}{2 \pi}} e^{-\frac{\tau}{2}(x-\mu)^2}
has R code expression: sqrt(tau/(2*pi))*exp(-tau/2*(x-mu)^2)

PMF of Hypergeometric 1
type: probability mass function
has name: PMF of Hypergeometric 1
has Latex expression: {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}
has R code expression: choose(K,k)*choose(M-K,n-k)/choose(M,n)

PDF of Pareto Type I
type: probability density function
has name: PDF of Pareto Type I
has Latex expression: \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m
has R code expression: (alpha * x_m^alpha) / x^(alpha+1)

relationship between Log-Normal 3 and Log-Normal 2 whereby \\mu = \\logm, v = \\sigma^2
type: Reparameterisation
distribution relationship from: Log-Normal 3
distribution relationship to: Log-Normal 2
distribution relationship definition in Latex: \mu = \log(m), v = \sigma^2
distribution relationship pair expression Latex: LogNormal3(m,\sigma) \rightarrow LogNormal2(\mu,v)
distribution relationship reference: ProbOnto spec

shape of Inverse-Gamma-1
type: mathematical object
has name: shape of Inverse-Gamma-1
has code name: InverseGamma1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha>0, \alpha \in R
has short name: shape
has short code name: shape

variance of Noncentral chi-squared 1
type: mathematical object
has name: variance of Noncentral chi-squared 1
has Latex expression: 2(n + 2 \delta)

skewness of Sinh-Arcsinh-2
type: mathematical object
has name: skewness of Sinh-Arcsinh-2
has code name: SinhArcsinh2.skewness
has Latex symbol: \epsilon
parameter has definition expressed in plain text: \epsilon \in R
has short name: skewness
has short code name: skewness

mean of Normal 3
type: mathematical object
has name: mean of Normal 3
has Latex expression: \mu

radius of Wigner-Semicircle-1
type: mathematical object
has name: radius of Wigner-Semicircle-1
has code name: WignerSemicircle1.radius
has Latex symbol: R
parameter has definition expressed in plain text: R > 0
has short name: radius
has short code name: radius

median of Exponential 1
type: mathematical object
has name: median of Exponential 1
has Latex expression: \lambda^{-1} \ln(2)

CDF of Hyperbolic secant 1
type: cumulative density function
has name: CDF of Hyperbolic secant 1
has Latex expression: \frac{\pi + 2 \text{ arctan}(\text{sinh}(\pi x))}{2\pi}
has R code expression: (pi + 2*arctan(sinh(pi*x)))/(2*pi)

median of Logistic 1
type: mathematical object
has name: median of Logistic 1
has Latex expression: \mu

mean of Generalized Poisson 3
type: mathematical object
has name: mean of Generalized Poisson 3
has Latex expression: \mu

LKJ Correlation 2
type: probability distribution
has name: LKJ Correlation 2
has code name: LKJCorrelation2
distribution has PDF: PDF of LKJ Correlation 2
has variate symbol in Latex: L
has Latex expression of support: \text{Cholesky factor of a symmetric positive-definite matrix with unit diagonal (i.e., a correlation matrix)}
distribution has parameter: shape of LKJ-Correlation-2

upper bound of Two-Sided-Power-1
type: mathematical object
has name: upper bound of Two-Sided-Power-1
has code name: TwoSidedPower1.upperBound
has Latex symbol: b
parameter has definition expressed in plain text: b \in R, b > m
has short name: upper bound
has short code name: upperBound

CDF of Chi 1
type: cumulative density function
has name: CDF of Chi 1
has Latex expression: \frac{ \gamma\left(\frac{k}{2},\,\frac{x^2}{2}\right) }{ \Gamma\left(\frac{k}{2}\right) }
has R code expression: Igamma(k/2,x^2/2,lower=T) / gamma(k/2)

Lower bound of Trapezoidal-1
type: mathematical object
has name: Lower bound of Trapezoidal-1
has code name: Trapezoidal1.lowerBound
has Latex symbol: a
parameter has definition expressed in plain text: a < d
has short name: Lower bound
has short code name: lowerBound

CDF of Normal 3
type: cumulative density function
has name: CDF of Normal 3
has Latex expression: \frac12\left[1 + \text{erf}\left( \frac{x-\mu}{\sqrt{1/\tau}\sqrt{2}}\right)\right]
has R code expression: 1/2*(1+erf((x-mu)/(sqrt(1/tau)*sqrt(2))))

Inverse Gaussian 1
type: probability distribution
label: Wald
has name: Inverse Gaussian 1
has code name: InverseGaussian1
distribution has PDF: PDF of Inverse Gaussian 1
distribution has CDF: CDF of Inverse Gaussian 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Inverse-Gaussian-1
distribution has parameter: mean of Inverse-Gaussian-1

success probability of Negative-Binomial-4
type: mathematical object
has name: success probability of Negative-Binomial-4
has code name: NegativeBinomial4.probability
has Latex symbol: p
parameter has definition expressed in plain text: p \in (0,1)
has short name: success probability
has short code name: probability

Pareto Type I
type: probability distribution
has name: Pareto Type I
has code name: ParetoTypeI1
distribution has PDF: PDF of Pareto Type I
distribution has CDF: CDF of Pareto Type I
distribution has mean: mean of Pareto Type I
distribution has median: median of Pareto Type I
distribution has mode: mode of Pareto Type I
distribution has variance: variance of Pareto Type I
has variate symbol in Latex: x
has Latex expression of support: x \in [x_m, +\infty)
distribution has parameter: scale of Pareto-Type-I
distribution has parameter: shape of Pareto-Type-I

CDF of Hypergeometric 1
type: cumulative density function
has name: CDF of Hypergeometric 1
has Latex expression: 1-{{{n \choose {k+1}}{{N-n} \choose {K-k-1}}}\over {N \choose K}} \,_3F_2\!\!\left[\begin{array}{c}1,\ k+1-K,\ k+1-n \\ k+2,\ N+k+2-K-n\end{array};1\right]
has R code expression: cumsum(PMF)

success probability of Geometric-1
type: mathematical object
has name: success probability of Geometric-1
has code name: Geometric1.probability
has Latex symbol: p
parameter has definition expressed in plain text: 0 < p < 1
has short name: success probability
has short code name: probability

relationship between Log-Normal 5 and Log-Normal 6 whereby m=\\exp\\mu, \\sigma_g= \\exp1/\\sqrt{\\tau}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m=\exp(\mu), \sigma_g= \exp(1/\sqrt{\tau})
distribution relationship pair expression Latex: LogNormal5(\mu,\tau) \rightarrow LogNormal6(m,\sigma_g)
distribution relationship reference: ProbOnto spec

Makeham 1
type: probability distribution
has name: Makeham 1
has code name: Makeham1
distribution has PDF: PDF of Makeham 1
distribution has CDF: CDF of Makeham 1
distribution has SF: SF of Makeham 1
distribution has HF: HF of Makeham 1
distribution has mean: mean of Makeham 1
distribution has median: median of Makeham 1
distribution has mode: mode of Makeham 1
distribution has variance: variance of Makeham 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: NA of Makeham-1

Benford 1
type: probability distribution
has name: Benford 1
has code name: Benford1
distribution has PMF: PMF of Benford 1
distribution has CDF: CDF of Benford 1
distribution has SF: SF of Benford 1
distribution has HF: HF of Benford 1
distribution has mean: mean of Benford 1
distribution has median: median of Benford 1
distribution has mode: mode of Benford 1
distribution has variance: variance of Benford 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{1,2,...,9\}

median of Normal 3
type: mathematical object
has name: median of Normal 3
has Latex expression: \mu

PMF of Zipf 1
type: probability mass function
has name: PMF of Zipf 1
has Latex expression: \frac{1}{x^\alpha H_{n,\alpha}} \\ H_{n,\alpha} = \sum^n_{i=1} (1/i)^\alpha
has R code expression: 1 / (x^alpha * Hfunc(n,alpha))\\ Hfunc = function(n,alpha) { Hsum=0; for(i in 1:n) { Hsum = Hsum + (1/i)^alpha } return(Hsum) }

mean of Exponential 1
type: mathematical object
has name: mean of Exponential 1
has Latex expression: \lambda^{-1}

variance of Generalized Gamma 2
type: mathematical object
has name: variance of Generalized Gamma 2
has Latex expression: b^2\{\Gamma(c+2/k)/\Gamma(c)-[\Gamma(c+1/k)/\Gamma(c)]^2\}

dispersion of Generalized-Poisson-3
type: mathematical object
has name: dispersion of Generalized-Poisson-3
has code name: GeneralizedPoisson3.dispersion
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > -1, \alpha \in R
has short name: dispersion
has short code name: dispersion

scale of Inverse-Gamma-1
type: mathematical object
has name: scale of Inverse-Gamma-1
has code name: InverseGamma1.scale
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta>0, \beta \in R
has short name: scale
has short code name: scale

median of Standard Power 1
type: mathematical object
has name: median of Standard Power 1
has Latex expression: (1/2)^{1/\beta}

shape paremeter of Amoroso-1
type: mathematical object
has name: shape paremeter of Amoroso-1
has code name: Amoroso1.shape2
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta \in R
has short name: shape paremeter
has short code name: shape2

mean of Hyperbolic secant 1
type: mathematical object
has name: mean of Hyperbolic secant 1
has Latex expression: 0

mean of Logistic 1
type: mathematical object
has name: mean of Logistic 1
has Latex expression: \mu

mean of Chi 1
type: mathematical object
has name: mean of Chi 1
has Latex expression: \mu = \sqrt{2}\,\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}

shape of Inverse-Gaussian-1
type: mathematical object
has name: shape of Inverse-Gaussian-1
has code name: InverseGaussian1.shape
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0
has short name: shape
has short code name: shape

Zero-Inflated Generalized Poisson 1
type: probability distribution
has name: Zero-Inflated Generalized Poisson 1
has code name: ZeroInflatedGeneralizedPoisson1
distribution has PMF: PMF of Zero-Inflated Generalized Poisson 1
distribution has CDF: CDF of Zero-Inflated Generalized Poisson 1
distribution has mean: mean of Zero-Inflated Generalized Poisson 1
distribution has variance: variance of Zero-Inflated Generalized Poisson 1
has variate symbol in Latex: y
has Latex expression of support: y \in \{0,1,2,3,\dots\}
distribution has parameter: mean of Zero-Inflated-Generalized-Poisson-1
distribution has parameter: probability of zero of Zero-Inflated-Generalized-Poisson-1
distribution has parameter: dispersion of Zero-Inflated-Generalized-Poisson-1

relationship between Laplace 1 and Laplace 2 whereby \\tau=1/b
type: Reparameterisation
distribution relationship from: Double Exponential 1
distribution relationship to: Double Exponential 2
distribution relationship definition in Latex: \tau=1/b
distribution relationship pair expression Latex: Laplace1(\mu,b) \rightarrow Laplace2(\mu,\tau)
distribution relationship reference: ProbOnto spec

mean of Pareto Type I
type: mathematical object
has name: mean of Pareto Type I
has Latex expression: \begin{cases} \infty & \text{for }\alpha\le 1 \\ \frac{\alpha\,x_m}{\alpha-1} & \text{for }\alpha>1 \end{cases}

mean of Hypergeometric 1
type: mathematical object
has name: mean of Hypergeometric 1
has Latex expression: n {K\over N}

mode of Normal 3
type: mathematical object
has name: mode of Normal 3
has Latex expression: \mu

SF of Exponential 1
type: survival function
has name: SF of Exponential 1
has Latex expression: \exp(-\lambda x)
has R code expression: exp(-lambda*x)

mode of Generalized Gamma 2
type: mathematical object
has name: mode of Generalized Gamma 2
has Latex expression: a+b(c-1/k)^{1/k}, c>1/k

degrees of freedom of Noncentral-chi-squared-1
type: mathematical object
has name: degrees of freedom of Noncentral-chi-squared-1
has code name: NoncentralChiSquared1.degreesOfFreedom
has Latex symbol: n
parameter has definition expressed in plain text: n \in N
has short name: degrees of freedom
has short code name: degreesOfFreedom

relationship between Log-Logistic 1 and Log-Logistic 2 whereby \\lambda=1/\\alpha, \\kappa=\\beta
type: Reparameterisation
distribution relationship from: Log-Logistic 1
distribution relationship to: Fisk
distribution relationship definition in Latex: \lambda=1/\alpha, \kappa=\beta
distribution relationship pair expression Latex: LogLogistic1(\alpha,\beta) \rightarrow LogLogistic2(\lambda,\kappa)
distribution relationship reference: ProbOnto spec

variance of Standard Power 1
type: mathematical object
has name: variance of Standard Power 1
has Latex expression: \frac{\beta}{(2+\beta)(1+\beta)^2}

mean of Generalized-Poisson-3
type: mathematical object
has name: mean of Generalized-Poisson-3
has code name: GeneralizedPoisson3.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu > 0
has short name: mean
has short code name: mean

Zipf 1
type: probability distribution
has name: Zipf 1
has code name: Zipf1
distribution has PMF: PMF of Zipf 1
distribution has CDF: CDF of Zipf 1
distribution has mean: mean of Zipf 1
distribution has variance: variance of Zipf 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{1,2,3,\dots\}
distribution has parameter: number of elements of Zipf-1
distribution has parameter: shape of Zipf-1

shape parameter of Amoroso-1
type: mathematical object
has name: shape parameter of Amoroso-1
has code name: Amoroso1.shape1
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R, \alpha > 0
has short name: shape parameter
has short code name: shape1

PDF of Makeham 1
type: probability density function
has name: PDF of Makeham 1
has Latex expression: (\gamma+\delta\kappa^x) e^{-\gamma x -\delta (\kappa^x - 1)/\log(\kappa)}
has R code expression: (gamma+delta*kappa^x)*exp(-gamma*x -delta*(kappa^x - 1)/log(kappa) )

median of Hyperbolic secant 1
type: mathematical object
has name: median of Hyperbolic secant 1
has Latex expression: 0

CDF of Inverse Gaussian 1
type: cumulative density function
has name: CDF of Inverse Gaussian 1
has Latex expression: \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu}-1 \right)\right) +\exp\left(\frac{2 \lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}}\left(\frac{x}{\mu}+1 \right)\right)
has R code expression: pnorm(sqrt(lambda/x) * (x/mu-1)) + exp(2*lambda/mu) * pnorm(-sqrt(lambda/x) * (x/mu+1))

Bernoulli 1
type: probability distribution
has name: Bernoulli 1
has code name: Bernoulli1
distribution has PMF: PMF of Bernoulli 1
distribution has CDF: CDF of Bernoulli 1
distribution has mean: mean of Bernoulli 1
distribution has median: median of Bernoulli 1
distribution has mode: mode of Bernoulli 1
distribution has variance: variance of Bernoulli 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1\}
distribution has parameter: probability of success of Bernoulli-1

mean of Trapezoidal 1
type: mathematical object
has name: mean of Trapezoidal 1
has Latex expression: \dfrac{2}{6(d+c-a-b)} \big(d^2+cd+c^2-b^2-ab-a^2)

mode of Chi 1
type: mathematical object
has name: mode of Chi 1
has Latex expression: \sqrt{k-1} \text{ for }k\ge 1

CDF of Pareto Type I
type: cumulative density function
has name: CDF of Pareto Type I
has Latex expression: 1-\left(\frac{x_m}{x}\right)^\alpha \text{ for } x \ge x_m
has R code expression: 1-(x_m/x)^alpha

variance of Power 1
type: mathematical object
has name: variance of Power 1
has Latex expression: \frac{\alpha^2 \beta}{(2+\beta)(1+\beta)^2}

CDF of Multivariate Normal 2
type: cumulative density function
has name: CDF of Multivariate Normal 2
has Latex expression: \text{no analytic expression}

CDF of Makeham 1
type: cumulative density function
has name: CDF of Makeham 1
has Latex expression: 1 - e^{-(\gamma x \log(\kappa) +\delta \kappa^x -\delta)/\log(\kappa)}
has R code expression: 1 - exp( -(gamma*x*log(kappa)+delta*kappa^x -delta)/log(kappa) )

PDF of Log-Normal 1
type: probability density function
has name: PDF of Log-Normal 1
has Latex expression: \frac{1}{x\sigma\sqrt{2\pi}}\ e^{-\frac{\left(\log x-\mu\right)^2}{2\sigma^2}}
has R code expression: 1/(x*sigma*sqrt(2*pi)) * exp((-(log(x)-mu)^2)/(2*sigma^2))

PDF of Multivariate Normal 3
type: probability density function
has name: PDF of Multivariate Normal 3
has Latex expression: (2\pi)^{-\frac{k}{2}}(LL^T)^{-\frac{1}{2}}\, e^{ -\frac{1}{2}(x-\mu)^T(LL^T)^{-1}(x-\mu) }

scale of Sinh-Arcsinh-1
type: mathematical object
has name: scale of Sinh-Arcsinh-1
has code name: SinhArcsinh1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: scale
has short code name: scale

CDF of Benford 1
type: cumulative density function
has name: CDF of Benford 1
has Latex expression: \log_{10}\left(1+x\right)
has R code expression: log10(1+x)

mode of Hyperbolic secant 1
type: mathematical object
has name: mode of Hyperbolic secant 1
has Latex expression: 0

shape parameter of Johnson-SU-1
type: mathematical object
has name: shape parameter of Johnson-SU-1
has code name: JohnsonSU1.shape1
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma \in R
has short name: shape parameter
has short code name: shape1

Generalized Negative Binomial 1
type: probability distribution
has name: Generalized Negative Binomial 1
has code name: GeneralizedNegativeBinomial1
distribution has PMF: PMF of Generalized Negative Binomial 1
distribution has CDF: CDF of Generalized Negative Binomial 1
distribution has mean: mean of Generalized Negative Binomial 1
distribution has variance: variance of Generalized Negative Binomial 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{0,1,2,3,\dots\}
distribution has parameter: shape2 of Generalized-Negative-Binomial-1
distribution has parameter: scale of Generalized-Negative-Binomial-1
distribution has parameter: shape1 of Generalized-Negative-Binomial-1

relationship between Generalized Gamma 2 and Exponential 1 whereby k=c=1, a=0, b=1/\\lambda
type: Reparameterisation
distribution relationship from: Generalized Gamma 2
distribution relationship to: Exponential 1
distribution relationship definition in Latex: k=c=1, a=0, b=1/\lambda
distribution relationship pair expression Latex: GeneralizedGamma2(a,b,c,k) \rightarrow Exponential1(\lambda)
distribution relationship reference: \cite{forbes2011statistical}

median of Geometric 1
type: mathematical object
has name: median of Geometric 1
has Latex expression: \left\lceil \frac{-1}{\log_2(1-p)}-1 \right\rceil \text{ (not unique if } -1/\log_2(1-p)-1 \text{ is an integer)}

left hand tail of Sinh-Arcsinh-1
type: mathematical object
has name: left hand tail of Sinh-Arcsinh-1
has code name: SinhArcsinh1.skewness
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu > 0
has short name: left hand tail
has short code name: skewness

median of Exponential 2
type: mathematical object
has name: median of Exponential 2
has Latex expression: \lambda^{-1} \ln(2)

shape of Weibull-2
type: mathematical object
has name: shape of Weibull-2
has code name: Weibull2.shape
has Latex symbol: v
has short name: shape
has short code name: shape

relationship between Log-Normal 6 and Log-Normal 4 whereby m=m, cv=\\sqrt{\\exp\\!\\big\\log^2\\sigma_g\\big-1}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m=m, cv=\sqrt{\exp\!\big(\log^2(\sigma_g)\big)-1}
distribution relationship pair expression Latex: LogNormal6(m,\sigma_g) \rightarrow LogNormal4(m,cv)
distribution relationship reference: ProbOnto spec

mean of Two-Sided Power 1
type: mathematical object
has name: mean of Two-Sided Power 1
has Latex expression: \frac{b-m+mn+a}{n+1}

CDF of Negative Binomial 4
type: cumulative density function
has name: CDF of Negative Binomial 4
has Latex expression: 1 - I_{p}(k+1,r)
has R code expression: 1 - Rbeta(p, k+1, r,lower = T)

PDF of Sinh-Arcsinh 2
type: probability density function
has name: PDF of Sinh-Arcsinh 2
has Latex expression: PDF(x;\mu,\sigma,\epsilon,\delta) = \phi[H(x;\mu,\sigma,\epsilon,\delta)] \, h(x;\mu,\sigma,\epsilon,\delta)\\ h(x;\mu,\sigma,\epsilon,\delta) = \frac{\delta\cosh\left(\delta\,\mbox{arcsinh}\left(\dfrac{x-\mu}{\sigma}\right)-\epsilon\right)}{\sigma \sqrt{1+\left(\dfrac{x-\mu}{\sigma}\right)^2}}\\ H(x;\mu,\sigma,\epsilon,\delta) = \sinh\left(\delta\,\mbox{arcsinh}\left(\dfrac{x-\mu}{\sigma}\right)-\epsilon\right)\\ \phi = \frac1{\sqrt{2\pi}} \exp(-x^2/2)
has R code expression: PDF = function(x,mu,sigma,epsilon,delta) { phi(Hfunc(x,mu,sigma,epsilon,delta)) * hfunc(x,mu,sigma,epsilon,delta) } hfunc = function(x,mu,sigma,epsilon,delta) { (delta*cosh(delta*asinh((x-mu)/sigma)-epsilon))/(sigma*sqrt(1+((x-mu)/sigma)^2)) } Hfunc = function(x,mu,sigma,epsilon,delta) { sinh(delta*asinh((x-mu)/sigma)-epsilon) } phi = function(x) { 1/(sqrt(2*pi))*exp(-x^2/2) }

variance of Binomial 1
type: mathematical object
has name: variance of Binomial 1
has Latex expression: np(1 - p)

mode of YuleSimon1
type: mathematical object
has name: mode of YuleSimon1
has Latex expression: 1

relationship between Standard Uniform 1 and Exponential 1 whereby -\\frac{1}{\\lambda} \\logX
type: Transformation
distribution relationship from: Standard Uniform 1
distribution relationship to: Exponential 1
distribution relationship definition in Latex: -\frac{1}{\lambda} \log(X)
distribution relationship pair expression Latex: StandardUniform1(0,1) \rightarrow Exponential1(\lambda)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformExponentialB.pdf}

relationship between Gamma 1 and Beta 1 whereby X1, X2 \\sim Gamma1k,\\theta \\text{ and } Y = X1/X1+X2 \\Rightarrow Y \\sim Beta1\\alpha,\\beta
type: Transformation
distribution relationship from: Gamma 1
distribution relationship to: Beta 1
distribution relationship definition in Latex: X1, X2 \sim Gamma1(k,\theta) \text{ and } Y = X1/(X1+X2) \Rightarrow Y \sim Beta1(\alpha,\beta)
distribution relationship pair expression Latex: Gamma1(k,\theta) \rightarrow Beta1(\alpha,\beta)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaBeta.pdf}

PDF of Multivariate Normal 2
type: probability density function
has name: PDF of Multivariate Normal 2
has Latex expression: (2\pi)^{-d/2}|T|^{\frac{1}{2}}\, \exp\big( -\frac{1}{2}(x-\mu)' T (x-\mu) \big)

HF of Makeham 1
type: hazard function
has name: HF of Makeham 1
has Latex expression: \gamma + \delta \kappa^x
has R code expression: gamma + delta*kappa^x

scale parameter of Power-1
type: mathematical object
has name: scale parameter of Power-1
has code name: Power1.scale
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: scale parameter
has short code name: scale

Multivariate Normal 3
type: probability distribution
label: Cholesky Parameterization
label: Multivariate Normal Distribution
has name: Multivariate Normal 3
has code name: MultivariateNormal3
distribution has PDF: PDF of Multivariate Normal 3
distribution has mean: mean of Multivariate Normal 3
distribution has mode: mode of Multivariate Normal 3
distribution has variance: variance of Multivariate Normal 3
has variate symbol in Latex: x
has Latex expression of support: x \in R^K
distribution has parameter: location of Multivariate-Normal-3
distribution has parameter: lower triangular matrix of Multivariate-Normal-3

Log-Normal 1
type: probability distribution
label: lognormal
label: Galton
has name: Log-Normal 1
has code name: LogNormal1
distribution has PDF: PDF of Log-Normal 1
distribution has CDF: CDF of Log-Normal 1
distribution has mean: mean of Log-Normal 1
distribution has median: median of Log-Normal 1
distribution has mode: mode of Log-Normal 1
distribution has variance: variance of Log-Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: mean of logx of Log-Normal-1
distribution has parameter: shape of Log-Normal-1

Beta-binomial 1
type: probability distribution
has name: Beta-binomial 1
has code name: BetaBinomial1
distribution has PMF: PMF of Beta-binomial 1
distribution has CDF: CDF of Beta-binomial 1
distribution has mean: mean of Beta-binomial 1
distribution has variance: variance of Beta-binomial 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,\dots,n\}
distribution has parameter: left beta parameter of Beta-binomial-1
distribution has parameter: right beta parameter of Beta-binomial-1
distribution has parameter: number of trials of Beta-binomial-1

PMF of Benford 1
type: probability mass function
has name: PMF of Benford 1
has Latex expression: \log_{10}\left(1+\frac1x\right)
has R code expression: log10(1+1/x)

shape parameter of Johnson-SU-1
type: mathematical object
has name: shape parameter of Johnson-SU-1
has code name: JohnsonSU1.shape2
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short name: shape parameter
has short code name: shape2

variance of Hyperbolic secant 1
type: mathematical object
has name: variance of Hyperbolic secant 1
has Latex expression: 1

lambda of Weibull-2
type: mathematical object
has name: lambda of Weibull-2
has code name: Weibull2.lambda
has Latex symbol: \lambda
has short name: lambda
has short code name: lambda

PMF of Generalized Negative Binomial 1
type: probability mass function
has name: PMF of Generalized Negative Binomial 1
has Latex expression: \frac{m}{m+\beta x} {m+\beta x \choose x} \theta^x (1-\theta)^{m+\beta x-x}
has R code expression: m/(m+beta*x) * choose(m+beta*x,x)*theta^x * (1-theta)^(m+beta*x-x)

mean of Exponential 2
type: mathematical object
has name: mean of Exponential 2
has Latex expression: \beta

relationship between Zero-Inflated Generalized Poisson 1 and Zero-inflated Poisson 1 whereby \\alpha=0, \\lambda=\\mu
type: Reparameterisation
distribution relationship from: Zero-Inflated Generalized Poisson 1
distribution relationship to: Zero-inflated Poisson 1
distribution relationship definition in Latex: \alpha=0, \lambda=\mu
distribution relationship pair expression Latex: ZeroInflatedGeneralizedPoisson1(\mu,\alpha,p0) \rightarrow ZeroInflatedPoisson1(\lambda,\pi)
distribution relationship reference: \cite{famoye2006zero}

relationship between Negative Binomial 2 and Negative Binomial 5 whereby \\alpha = 1/\\tau, \\beta = 1 / \\tau \\lambda
type: Reparameterisation
distribution relationship from: Negative Binomial 2
distribution relationship to: Negative Binomial 5
distribution relationship definition in Latex: \alpha = 1/\tau, \beta = 1 / (\tau \lambda)
distribution relationship pair expression Latex: NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial5(\alpha, \beta)
distribution relationship reference: ProbOnto spec

mean of Geometric 1
type: mathematical object
has name: mean of Geometric 1
has Latex expression: \frac{1-p}{p}

mean of Negative Binomial 4
type: mathematical object
has name: mean of Negative Binomial 4
has Latex expression: \frac{pr}{1-p}

SF of Two-Sided Power 1
type: survival function
has name: SF of Two-Sided Power 1
has Latex expression: \begin{cases} -\frac{-b+a+(x-a)^n (m-a)^{1-n}}{b-a} &\text{ for } a < x < m \\ \frac{(b-x)^n (b-m)^{1-n}}{b-a} &\text{ for } m \leq x < b \end{cases}
has R code expression: SF1 = function(x,a,b,m,n) { -(-b+a+(x-a)^n*(m-a)^(1-n))/(b-a) } SF2 = function(x,a,b,m,n) { ((b-x)^n*(b-m)^(1-n))/(b-a) }

variance of YuleSimon1
type: mathematical object
has name: variance of YuleSimon1
has Latex expression: \frac{\rho^2}{(\rho-1)^2(\rho-2)} \text{ for } \rho>2

relationship between Negative Binomial 3 and Negative Binomial 4 whereby r = \\phi, p = \\mu / \\phi + \\mu
type: Reparameterisation
distribution relationship from: Negative Binomial 3
distribution relationship to: Negative Binomial 4
distribution relationship definition in Latex: r = \phi, p = \mu / (\phi + \mu)
distribution relationship pair expression Latex: NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial4(r,p)
distribution relationship reference: ProbOnto spec

number of trials of Binomial-1
type: mathematical object
has name: number of trials of Binomial-1
has code name: Binomial1.numberOfTrials
has Latex symbol: n
parameter has definition expressed in plain text: n \in N, n \ge 0
has short name: number of trials
has short code name: numberOfTrials

relationship between Negative Binomial 4 and Negative Binomial 2 whereby \\tau=1/r, \\lambda = \\frac{rp}{1-p}
type: Reparameterisation
distribution relationship from: Negative Binomial 4
distribution relationship to: Negative Binomial 2
distribution relationship definition in Latex: \tau=1/r, \lambda = \frac{rp}{1-p}
distribution relationship pair expression Latex: NegativeBinomial4(r,p) \rightarrow NegativeBinomial2(\lambda, \tau)
distribution relationship reference: ProbOnto spec \\ \cite{cameron2013regression}

shape parameter of Power-1
type: mathematical object
has name: shape parameter of Power-1
has code name: Power1.shape
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape parameter
has short code name: shape

Multivariate Normal 2
type: probability distribution
label: Multivariate Gaussian 2
has name: Multivariate Normal 2
has code name: MultivariateNormal2
distribution has PDF: PDF of Multivariate Normal 2
distribution has CDF: CDF of Multivariate Normal 2
distribution has mean: mean of Multivariate Normal 2
distribution has mode: mode of Multivariate Normal 2
distribution has variance: variance of Multivariate Normal 2
has variate symbol in Latex: x
has Latex expression of support: x \in R^k
distribution has parameter: precision matrix of Multivariate-Normal-2
distribution has parameter: location of Multivariate-Normal-2

SF of Makeham 1
type: survival function
has name: SF of Makeham 1
has Latex expression: e^{-(\gamma x \log(\kappa) +\delta \kappa^x -\delta)/\log(\kappa)}
has R code expression: exp( -(gamma*x*log(kappa)+delta*kappa^x -delta)/log(kappa) )

SF of Benford 1
type: survival function
has name: SF of Benford 1
has Latex expression: 1-\log_{10}(x)
has R code expression: 1-log10(x)

mean of Multivariate Normal 3
type: mathematical object
has name: mean of Multivariate Normal 3
has Latex expression: \mu

Negative Binomial 2
type: probability distribution
has name: Negative Binomial 2
has code name: NegativeBinomial2
distribution has PMF: PMF of Negative Binomial 2
distribution has CDF: CDF of Negative Binomial 2
distribution has mean: mean of Negative Binomial 2
distribution has variance: variance of Negative Binomial 2
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: Poisson intensity of Negative-Binomial-2
distribution has parameter: overdispersion of Negative-Binomial-2

Ordered Logistic 1
type: probability distribution
has name: Ordered Logistic 1
has code name: OrderedLogistic1
distribution has PMF: PMF of Ordered Logistic 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{1,\dots, K\}
distribution has parameter: predictor of Ordered-Logistic-1
distribution has parameter: ordered cutpoints of Ordered-Logistic-1

PMF of Beta-binomial 1
type: probability mass function
has name: PMF of Beta-binomial 1
has Latex expression: {n \choose k}\frac{B(k+\alpha,n-k+\beta)}{B(\alpha,\beta)}
has R code expression: choose(n,k) * beta(k+alpha,n-k+beta) / beta(alpha,beta)

rate or inverse scale of Exponential-1
type: mathematical object
has name: rate or inverse scale of Exponential-1
has code name: Exponential1.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0
has short name: rate or inverse scale
has short code name: rate

scale of Sinh-Arcsinh-2
type: mathematical object
has name: scale of Sinh-Arcsinh-2
has code name: SinhArcsinh2.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: scale
has short code name: scale

SF of Exponential 2
type: survival function
has name: SF of Exponential 2
has Latex expression: e^{-x/\beta}
has R code expression: exp(-x/beta)

location of Sinh-Arcsinh-2
type: mathematical object
has name: location of Sinh-Arcsinh-2
has code name: SinhArcsinh2.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

scale of Logit-Normal-1
type: mathematical object
has name: scale of Logit-Normal-1
has code name: LogitNormal1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0, \sigma \in R
has short name: scale
has short code name: scale

Lower bound of Two-Sided-Power-1
type: mathematical object
has name: Lower bound of Two-Sided-Power-1
has code name: TwoSidedPower1.lowerBound
has Latex symbol: a
parameter has definition expressed in plain text: a \in R
has short name: Lower bound
has short code name: lowerBound

success probability in each trial of Binomial-1
type: mathematical object
has name: success probability in each trial of Binomial-1
has code name: Binomial1.probability
has Latex symbol: p
parameter has definition expressed in plain text: p \in [0,1]
has short name: success probability in each trial
has short code name: probability

Zeta 1
type: probability distribution
has name: Zeta 1
has code name: Zeta1
distribution has PMF: PMF of Zeta 1
distribution has CDF: CDF of Zeta 1
distribution has mean: mean of Zeta 1
distribution has variance: variance of Zeta 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{1,2,3,\dots\}
distribution has parameter: shape of Zeta-1

PROB k0000953
type: Transformation
label: relationship between Standard Uniform 1 and Log-Logistic 2 whereby \\text{If } X \\sim StandardUniform1 \\text{ and } Y=\\frac{1}{\\lambda} \\Big \\frac{1-X}{X} \\Big^{1/\\kappa} \\Rightarrow Y\\sim LogLogistic2\\lambda,\\kappa
distribution relationship from: Standard Uniform 1
distribution relationship to: Fisk
distribution relationship definition in Latex: \text{If } X \sim StandardUniform1 \text{ and } Y=\frac{1}{\lambda} \Big( \frac{1-X}{X} \Big)^{1/\kappa} \Rightarrow Y\sim LogLogistic2(\lambda,\kappa)
distribution relationship pair expression Latex: StandardUniform1 \rightarrow LogLogistic2(\lambda,\kappa)

PMF of Bernoulli 2
type: probability mass function
has name: PMF of Bernoulli 2
has Latex expression: \begin{cases} logit^{-1}(\alpha) & \text{for }k=1 \\ 1 - logit^{-1}(\alpha) & \text{for }k=0 \end{cases}
has R code expression: invLogit = function(x) { exp(x)/(1+exp(x)) } invLogit(alpha1) for k = 1 1-invLogit(alpha1) for k = 0

mode of Negative Binomial 4
type: mathematical object
has name: mode of Negative Binomial 4
has Latex expression: \begin{cases} \lfloor \frac{p(r-1)}{1-p} \rfloor & \text{for } r > 1 \\ 0 & \text{for } r \leq 1 \end{cases}

shape parameter of Burr-1
type: mathematical object
has name: shape parameter of Burr-1
has code name: Burr1.shape2
has Latex symbol: k
parameter has definition expressed in plain text: k > 0
has short name: shape parameter
has short code name: shape2

relationship between Skew Normal and Standard Normal 1 whereby \\text{If } X \\sim SkewNormal1\\mu,\\sigma,0 \\text{ then } X \\sim StandardNormal1
type: Special case
distribution relationship from: Skew Normal
distribution relationship to: Standard Normal 1
distribution relationship definition in Latex: \text{If } X \sim SkewNormal1(\mu,\sigma,0) \text{ then } X \sim StandardNormal1
distribution relationship pair expression Latex: SkewNormal1(\mu,\sigma,\alpha) \rightarrow StandardNormal1
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Skew_normal_distribution}\\ \url{http://azzalini.stat.unipd.it/SN/Intro/intro.html}

variance of Geometric 1
type: mathematical object
has name: variance of Geometric 1
has Latex expression: \frac{1-p}{p^2}

relationship between Generalized Gamma 1 and Generalized Gamma 3 whereby r=d/p, \\beta=p, \\mu = 1/a
type: Reparameterisation
distribution relationship from: Generalized Gamma 1
distribution relationship to: Generalized Gamma 3
distribution relationship definition in Latex: r=d/p, \beta=p, \mu = 1/a
distribution relationship pair expression Latex: GeneralizedGamma1(a,d,p) \rightarrow GeneralizedGamma3(r,\mu,\beta)
distribution relationship reference: ProbOnto spec

relationship between Generalized Gamma 1 and Gamma 1 whereby p=1, k=d, \\theta=a
type: Special case
distribution relationship from: Generalized Gamma 1
distribution relationship to: Gamma 1
distribution relationship definition in Latex: p=1, k=d, \theta=a
distribution relationship pair expression Latex: GeneralizedGamma1(a,d,p) \rightarrow Gamma1(k,\theta)
distribution relationship reference: ProbOnto spec

shape parameter of YuleSimon1
type: mathematical object
has name: shape parameter of YuleSimon1
has code name: YuleSimon1.shape
has Latex symbol: \rho
parameter has definition expressed in plain text: \rho > 0
has short name: shape parameter
has short code name: shape

mean of Makeham 1
type: mathematical object
has name: mean of Makeham 1
has Latex expression: \text{mathematically intractable}

Maxwell Boltzmann 1
type: probability distribution
label: Maxwell
has name: Maxwell Boltzmann 1
has code name: MaxwellBoltzmann1
distribution has PDF: PDF of Maxwell Boltzmann 1
distribution has CDF: CDF of Maxwell Boltzmann 1
distribution has mean: mean of Maxwell Boltzmann 1
distribution has median: median of Maxwell Boltzmann 1
distribution has variance: variance of Maxwell Boltzmann 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: scale parameter of Maxwell-Boltzmann-1

HF of Benford 1
type: hazard function
has name: HF of Benford 1
has Latex expression: \frac{\log_{10}\left(1+\frac1x\right)}{1-\log_{10}x}
has R code expression: log10(1+1/x) / (1-log10(x))

dispersion of GeneralizedPoisson2
type: mathematical object
has name: dispersion of GeneralizedPoisson2
has code name: GeneralizedPoisson2.dispersion
has Latex symbol: \delta
parameter has definition expressed in plain text: max(-1,-\mu/m) < \delta < 1 \text{ with } m (\geq 4) \text { the largest positive integer for which } \mu + m \delta > 0.
has short name: dispersion
has short code name: dispersion

CDF of Beta-binomial 1
type: cumulative density function
has name: CDF of Beta-binomial 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

PMF of Negative Binomial 2
type: probability mass function
has name: PMF of Negative Binomial 2
has Latex expression: \frac{\Gamma(k+\frac{1}{\tau})}{k!\; \Gamma(\frac{1}{\tau})} \Big(\frac{1}{1+\tau \lambda} \Big)^{\frac{1}{\tau}} \Big(\frac{\lambda}{\frac{1}{\tau} + \lambda} \Big)^{k}
has R code expression: gamma(k + 1/tau)/(factorial(k) * gamma(1/tau)) * 1/(1+tau*lambda)^(1/tau) * (lambda/(1/tau + lambda))^k

variance of Exponential 1
type: mathematical object
has name: variance of Exponential 1
has Latex expression: \lambda^{-2}

CDF of Johnson SU 1
type: cumulative density function
has name: CDF of Johnson SU 1
has Latex expression: \frac12\left(1+ \text{erf}\left[\frac{\gamma + \delta \,\text{arcsin}\left[\frac{x-\mu}{\sigma}\right]}{\sqrt{2}}\right] \right)
has R code expression: 1/2 *(1+ erf((gamma+delta*asinh((x-mu)/sigma))/sqrt(2)))

relationship between Logit Normal 1 and Normal 1 whereby X \\text{ is logit-normally distributed, then } Y = logitX= \\logX/1-X \\text{ is normally distributed}
type: Transformation
distribution relationship from: Logit Normal 1
distribution relationship to: Normal
distribution relationship definition in Latex: X \text{ is logit-normally distributed, then } Y = logit(X)= \log(X/(1-X)) \text{ is normally distributed}
distribution relationship pair expression Latex: LogitNormal1(\mu,\sigma) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Logit-normal_distribution}

variance of Two-Sided Power 1
type: mathematical object
has name: variance of Two-Sided Power 1
has Latex expression: \frac{-2mnb +2m^2n+a^2n+b^2n-2mna+2bm-2ba+2am-2m^2}{(n+1)^2(n+2)}

Hypergeometric 1
type: probability distribution
has name: Hypergeometric 1
has code name: Hypergeometric1
distribution has PMF: PMF of Hypergeometric 1
distribution has CDF: CDF of Hypergeometric 1
distribution has mean: mean of Hypergeometric 1
distribution has mode: mode of Hypergeometric 1
distribution has variance: variance of Hypergeometric 1
has variate symbol in Latex: k
has Latex expression of support: k\, \in\, \left\{\max{(0,\, n+K-N)},\, \dots,\, \min{(n,\, K )}\right\}
distribution has parameter: number of successes of Hypergeometric-1
distribution has parameter: number of trials of Hypergeometric-1
distribution has parameter: population size of Hypergeometric-1

CDF of Sinh-Arcsinh 2
type: cumulative density function
has name: CDF of Sinh-Arcsinh 2
has Latex expression: S_0(x;\mu,\sigma,\epsilon,\delta) = \Phi[H(x;\mu,\sigma,\epsilon,\delta)]\\ \Phi(x) = \frac12 (1 + erf(x/\sqrt{2}))\\ H(x;\mu,\sigma,\epsilon,\delta) = \sinh\left(\delta\,\mbox{arcsinh}\left(\dfrac{x-\mu}{\sigma}\right)-\epsilon\right)
has R code expression: S0func = function(x,mu,sigma,epsilon,delta) { PHI(Hfunc(x,mu,sigma,epsilon,delta)) } Hfunc = function(x,mu,sigma,epsilon,delta) { sinh(delta*asinh((x-mu)/sigma)-epsilon) } PHI = function(x) { 1/2 * (1 + erf(x/(sqrt(2)))) }

HF of Exponential 2
type: hazard function
has name: HF of Exponential 2
has Latex expression: 1/\beta
has R code expression: 1/beta

variance of Negative Binomial 4
type: mathematical object
has name: variance of Negative Binomial 4
has Latex expression: \frac{pr}{(1-p)^2}

mode of Geometric 1
type: mathematical object
has name: mode of Geometric 1
has Latex expression: 0

relationship between Normal 1 and Log-Normal 1 whereby \\expX
type: Transformation
distribution relationship from: Normal
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \exp(X)
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalLognormal.pdf}\\ \cite{forbes2011statistical}

right hand tail of Sinh-Arcsinh-1
type: mathematical object
has name: right hand tail of Sinh-Arcsinh-1
has code name: SinhArcsinh1.kurtosis
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau > 0
has short name: right hand tail
has short code name: kurtosis

Bernoulli 2
type: probability distribution
has name: Bernoulli 2
has code name: Bernoulli2
distribution has PMF: PMF of Bernoulli 2
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1\}
distribution has parameter: logit of probability of success of Bernoulli-2

shape parameter of Burr-1
type: mathematical object
has name: shape parameter of Burr-1
has code name: Burr1.shape1
has Latex symbol: c
parameter has definition expressed in plain text: c > 0
has short name: shape parameter
has short code name: shape1

Gaussian
type: probability distribution
label: Normal
label: Normal 3
has name: Normal 3
has code name: Normal3
distribution has PDF: PDF of Normal 3
distribution has CDF: CDF of Normal 3
distribution has mean: mean of Normal 3
distribution has median: median of Normal 3
distribution has mode: mode of Normal 3
distribution has variance: variance of Normal 3
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: mean of Normal-3
distribution has parameter: precision of Normal-3

Chi 1
type: probability distribution
has name: Chi 1
has code name: Chi1
distribution has PDF: PDF of Chi 1
distribution has CDF: CDF of Chi 1
distribution has mean: mean of Chi 1
distribution has mode: mode of Chi 1
distribution has variance: variance of Chi 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: degrees of freedom of Chi-1

relationship between Negative Binomial 1 and Normal 1 whereby \\mu = n1-p, n \\rightarrow \\infty
type: Limiting
distribution relationship from: Negative Binomial 1
distribution relationship to: Normal
distribution relationship definition in Latex: \mu = n(1-p), n \rightarrow \infty
distribution relationship pair expression Latex: NegativeBinomial1(r,p) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalNormal.pdf}

relationship between Normal 3 and Normal 1 whereby \\mu = \\mu, \\sigma = 1 / \\sqrt{\\tau}
type: Reparameterisation
distribution relationship from: Gaussian
distribution relationship to: Normal
distribution relationship definition in Latex: \mu = \mu, \sigma = 1 / \sqrt{\tau}
distribution relationship pair expression Latex: Normal3(\mu,\tau) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

Inverse-Wishart 1
type: probability distribution
has name: Inverse-Wishart 1
has code name: InverseWishart1
distribution has PDF: PDF of Inverse-Wishart 1
distribution has mean: mean of Inverse-Wishart 1
distribution has mode: mode of Inverse-Wishart 1
has variate symbol in Latex: X
has Latex expression of support: X(p \times p) - \text{positive-definite matrix}
distribution has parameter: degrees of freedom of Inverse-Wishart-1
distribution has parameter: scale matrix of Inverse-Wishart-1

minimum of Standard-Uniform-1
type: mathematical object
has name: minimum of Standard-Uniform-1
has code name: StandardUniform1.minimum
has Latex symbol: a
parameter has definition expressed in plain text: a=0
has short name: minimum
has short code name: minimum

variance of Half Cauchy 1
type: mathematical object
has name: variance of Half Cauchy 1
has Latex expression: undefined

Geometric 1
type: probability distribution
has name: Geometric 1
has code name: Geometric1
distribution has PMF: PMF of Geometric 1
distribution has CDF: CDF of Geometric 1
distribution has SF: SF of Geometric 1
distribution has HF: HF of Geometric 1
distribution has mean: mean of Geometric 1
distribution has median: median of Geometric 1
distribution has mode: mode of Geometric 1
distribution has variance: variance of Geometric 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}, \text{ number of failures}
distribution has parameter: success probability of Geometric-1

median of Log-Logistic 1
type: mathematical object
has name: median of Log-Logistic 1
has Latex expression: \alpha

relationship between Negative Binomial 4 and Negative Binomial 3 whereby \\phi = r, \\mu = rp / 1 - p
type: Reparameterisation
distribution relationship from: Negative Binomial 4
distribution relationship to: Negative Binomial 3
distribution relationship definition in Latex: \phi = r, \mu = rp / (1 - p)
distribution relationship pair expression Latex: NegativeBinomial4(r,p) \rightarrow NegativeBinomial3(\mu, \phi)
distribution relationship reference: ProbOnto spec

relationship between Log-Normal 4 and Log-Normal 5 whereby \\mu = \\logm, \\tau = 1 / \\logcv^2 + 1
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu = \log(m), \tau = 1 / \log(cv^2 + 1)
distribution relationship pair expression Latex: LogNormal4(m,cv) \rightarrow LogNormal5(\mu,\tau)
distribution relationship reference: ProbOnto spec

PDF of Two-Sided Power 1
type: probability density function
has name: PDF of Two-Sided Power 1
has Latex expression: \begin{cases} \frac{n (x-a)^{n-1}}{(b-a)(m-a)^{n-1}} \text{ for } a < x < m \\ \frac{n (b-x)^{n-1}}{(b-a)(b-m)^{n-1}} \text{ for } m \leq x < b \end{cases}
has R code expression: PDF1 = function(x,a,b,m,n) { (n*(x-a)^(n-1))/((b-a)*(m-a)^(n-1)) } PDF2 = function(x,a,b,m,n) { (n*(b-x)^(n-1))/((b-a)*(b-m)^(n-1)) }

PDF of Log-Uniform 1
type: probability density function
has name: PDF of Log-Uniform 1
has Latex expression: \frac{1}{x(\log(max) - \log(min))}
has R code expression: 1/(x*(log(max) - log(min)))

CDF of Zeta 1
type: cumulative density function
has name: CDF of Zeta 1
has Latex expression: \frac{\sum^x_{i=1} (1/i)^\alpha}{\zeta(\alpha)}
has R code expression: Hfunc(x,alpha)/zeta(\alpha) Hfunc = function(x,alpha) { Hsum=array(0,length(x)); for(i in 1:length(x)) { Hsum[i] = (1/i)^alpha } return(cumsum(Hsum)) }

CDF of Exponential 2
type: cumulative density function
has name: CDF of Exponential 2
has Latex expression: 1 - \exp(-x/\beta)
has R code expression: 1 - exp(-x/beta)

CDF of Negative Binomial 2
type: cumulative density function
has name: CDF of Negative Binomial 2
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

PDF of Frechet 1
type: probability density function
has name: PDF of Frechet 1
has Latex expression: \frac{\alpha}{\sigma} \Big(\frac{x}{\sigma}\Big)^{-\alpha-1} \exp\Big(-\Big(\frac{x}{\sigma}\Big)^{-\alpha}\Big)
has R code expression: alpha/sigma * (x/sigma)^(-alpha-1) * exp(-(x/sigma)^(-alpha))

relationship between Weibull Discrete 1 and Geometric 1 whereby \\text{The Geometric1p distribution is a special case of the WeibullDiscrete1} p, \\beta \\text{ distribution when } \\beta = 1.
type: Special case
distribution relationship from: Weibull Discrete 1
distribution relationship to: Geometric 1
distribution relationship definition in Latex: \text{The Geometric1(p) distribution is a special case of the WeibullDiscrete1} (p, \beta) \text{ distribution when } \beta = 1.
distribution relationship pair expression Latex: WeibullDiscrete1(p\beta) \rightarrow Geometric1(p)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/DiscreteweibullGeometric.pdf}

mean of F 1
type: mathematical object
has name: mean of F 1
has Latex expression: n_2/(n_2-2), n_2>2

relationship between Log-Normal 7 and Log-Normal 4 whereby m = \\mu_N/\\sqrt{1+\\sigma_N^2/\\mu_N^2}, cv = \\sigma_N/\\mu_N
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m = \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}, cv = \sigma_N/\mu_N
distribution relationship pair expression Latex: LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal4(m,cv)
distribution relationship reference: ProbOnto spec

relationship between Rice 1 and Exponential 1 whereby \\text{If } R \\sim Rice10,\\sigma \\text{ then } R^2 \\sim Exponential1\\lambda
type: Transformation
distribution relationship from: Rician
distribution relationship to: Exponential 1
distribution relationship definition in Latex: \text{If } R \sim Rice1(0,\sigma) \text{ then } R^2 \sim Exponential1(\lambda)
distribution relationship pair expression Latex: Rice1(\nu,\sigma) \rightarrow Exponential1(\lambda)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Rice_distribution}

mean of Log-Logistic 1
type: mathematical object
has name: mean of Log-Logistic 1
has Latex expression: \frac{\alpha \pi/\beta}{\sin(\pi/\beta)} \text{ if } \beta>1, \text{ else undefined}

CDF of Log-Logistic 2
type: cumulative density function
has name: CDF of Log-Logistic 2
has Latex expression: \frac{(\lambda x)^\kappa}{1+(\lambda x)^\kappa}
has R code expression: (lambda*x)^kappa / (1+(lambda*x)^kappa)

median of Makeham 1
type: mathematical object
has name: median of Makeham 1
has Latex expression: \text{mathematically intractable}

variance of Arcsine 2
type: mathematical object
has name: variance of Arcsine 2
has Latex expression: (b-a)^2/8

PDF of Inverse-Wishart 1
type: probability density function
has name: PDF of Inverse-Wishart 1
has Latex expression: \frac{\left|\Psi\right|^{\frac{\nu}{2}}}{2^{\frac{\nu p}{2}}\Gamma_p(\frac{\nu}{2})} \left|X\right|^{-\frac{\nu+p+1}{2}}e^{-\frac{1}{2}\text{tr}(\Psi X^{-1})}

mode of Log-Logistic 1
type: mathematical object
has name: mode of Log-Logistic 1
has Latex expression: \alpha \Big(\frac{\beta-1}{\beta+1}\Big)^{1/\beta} \text{ if } \beta>1, 0 \text{ otherwise}

relationship between Exponential 1 and Exponential 2 whereby \\beta=1/\\lambda
type: Reparameterisation
distribution relationship from: Exponential 1
distribution relationship to: Negative exponential
distribution relationship definition in Latex: \beta=1/\lambda
distribution relationship pair expression Latex: Exponential1(\lambda) \rightarrow Exponential2(\beta)
distribution relationship reference: ProbOnto spec

maximum of Standard-Uniform-1
type: mathematical object
has name: maximum of Standard-Uniform-1
has code name: StandardUniform1.maximum
has Latex symbol: b
parameter has definition expressed in plain text: b=1
has short name: maximum
has short code name: maximum

median of Half Cauchy 1
type: mathematical object
has name: median of Half Cauchy 1
has Latex expression: \tan(\pi/4)

concentration of Von-Mises-1
type: mathematical object
has name: concentration of Von-Mises-1
has code name: VonMises1.concentration
has Latex symbol: \kappa
parameter has definition expressed in plain text: \kappa \in R^+
has short name: concentration
has short code name: concentration

CDF of Log-Uniform 1
type: cumulative density function
has name: CDF of Log-Uniform 1
has Latex expression: \frac{\log(x) - \log(min)}{\log(max) - \log(min)}
has R code expression: (log(x) - log(min)) / (log(max) - log(min))

Two-Sided Power 1
type: probability distribution
label: TSP
has name: Two-Sided Power 1
has code name: TwoSidedPower1
distribution has PDF: PDF of Two-Sided Power 1
distribution has CDF: CDF of Two-Sided Power 1
distribution has SF: SF of Two-Sided Power 1
distribution has HF: HF of Two-Sided Power 1
distribution has mean: mean of Two-Sided Power 1
distribution has variance: variance of Two-Sided Power 1
has variate symbol in Latex: x
has Latex expression of support: x \in (a,b)
distribution has parameter: shape of Two-Sided-Power-1
distribution has parameter: Lower bound of Two-Sided-Power-1
distribution has parameter: location of Two-Sided-Power-1
distribution has parameter: upper bound of Two-Sided-Power-1

PMF of Zeta 1
type: probability mass function
has name: PMF of Zeta 1
has Latex expression: \frac{1}{x^\alpha \zeta(\alpha)}
has R code expression: 1 / (x^alpha * zeta(alpha))

relationship between Double Poisson 1 and Poisson 1 whereby \\phi=1
type: Special case
distribution relationship from: Double Poisson 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \phi=1
distribution relationship pair expression Latex: DoublePoisson1(\mu,\phi) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{cameron2013regression}

Frechet 1
type: probability distribution
has name: Frechet 1
has code name: Frechet1
distribution has PDF: PDF of Frechet 1
distribution has CDF: CDF of Frechet 1
distribution has mean: mean of Frechet 1
distribution has median: median of Frechet 1
distribution has mode: mode of Frechet 1
distribution has variance: variance of Frechet 1
has variate symbol in Latex: x
has Latex expression of support: x \in R^+
distribution has parameter: scale of Frechet-1
distribution has parameter: shape of Frechet-1

PDF of Exponential 2
type: probability density function
has name: PDF of Exponential 2
has Latex expression: 1/\beta e^{-x/\beta}
has R code expression: 1/beta*exp(-1/beta*x)

relationship between Log-Normal 5 and Log-Normal 7 whereby \\mu_N = \\exp\\big\\mu + 1/2 \\tau\\big, \\sigma_N = \\exp\\big\\mu + 1/2 \\tau\\big \\sqrt{\\exp\\big1/\\tau\\big-1}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu_N = \exp\big(\mu + 1/(2 \tau)\big), \sigma_N = \exp\big(\mu + 1/(2 \tau)\big) \sqrt{\exp\big(1/\tau\big)-1}
distribution relationship pair expression Latex: LogNormal5(\mu,\tau) \rightarrow LogNormal7(\mu_N,\sigma_N)
distribution relationship reference: ProbOnto spec

mean of Negative Binomial 2
type: mathematical object
has name: mean of Negative Binomial 2
has Latex expression: \lambda

location of Normal-inverse-gamma-1
type: mathematical object
has name: location of Normal-inverse-gamma-1
has code name: NormalInverseGamma1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: mean

CDF of F 1
type: cumulative density function
has name: CDF of F 1
has Latex expression: I_{\frac{n_1 x}{n_1 x + n_2}} \left(\tfrac{n_1}{2}, \tfrac{n_2}{2}\right)
has R code expression: Rbeta(n1*x / (n1*x + n2), n1/2, n2/2)

relationship between Poisson 1 and Normal 1 whereby \\sigma^2 = \\lambda, \\mu = \\lambda, \\lambda \\rightarrow \\infty
type: Limiting
distribution relationship from: Poisson 1
distribution relationship to: Normal
distribution relationship definition in Latex: \sigma^2 = \lambda, \mu = \lambda, \lambda \rightarrow \infty
distribution relationship pair expression Latex: Poisson1(\lambda) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PoissonNormal.pdf}

relationship between Student's t-distribution 2 and Student's t-distribution 3 whereby \\sigma=1/\\sqrt{\\tau}
type: Reparameterisation
distribution relationship from: t-distribution
distribution relationship to: non-standardized Student's t-distribution
distribution relationship definition in Latex: \sigma=1/\sqrt{\tau}
distribution relationship pair expression Latex: StudentT2(\mu,\tau,k) \rightarrow StudentT3(\nu,\mu,\sigma)
distribution relationship reference: ProbOnto spec

degrees of freedom of Noncentral-F-1
type: mathematical object
has name: degrees of freedom of Noncentral-F-1
has code name: NoncentralF1.degreesOfFreedom2
has Latex symbol: n
parameter has definition expressed in plain text: n \in N
has short name: degrees of freedom
has short code name: degreesOfFreedom2

mean of Log-Logistic 2
type: mathematical object
has name: mean of Log-Logistic 2
has Latex expression: \frac{\pi}{\kappa \lambda \sin(\pi/\kappa)}

lower bound of Arcsine-2
type: mathematical object
has name: lower bound of Arcsine-2
has code name: Arcsine2.lowerBound
has Latex symbol: a
parameter has definition expressed in plain text: a \in R
has short name: lower bound
has short code name: lowerBound

mode of Makeham 1
type: mathematical object
has name: mode of Makeham 1
has Latex expression: \text{mathematically intractable}

CDF of Geometric 1
type: cumulative density function
has name: CDF of Geometric 1
has Latex expression: 1-(1 - p)^{k+1}
has R code expression: 1-(1 - p)^(k+1)

variance of Log-Logistic 1
type: mathematical object
has name: variance of Log-Logistic 1
has Latex expression: \alpha^2 \Big(\frac{2\pi/\beta}{\sin(2\pi/\beta)} - \frac{(\pi/\beta)^2}{\sin^2(\pi/\beta)} \Big), \text{ for } \beta>2

mean of Half Cauchy 1
type: mathematical object
has name: mean of Half Cauchy 1
has Latex expression: undefined

Gamma 2
type: probability distribution
has name: Gamma 2
has code name: Gamma2
distribution has PDF: PDF of Gamma 2
distribution has CDF: CDF of Gamma 2
distribution has mean: mean of Gamma 2
distribution has variance: variance of Gamma 2
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Gamma-2
distribution has parameter: rate of Gamma-2

Negative exponential
type: probability distribution
label: Exponential 2
has name: Exponential 2
has code name: Exponential2
distribution has PDF: PDF of Exponential 2
distribution has CDF: CDF of Exponential 2
distribution has SF: SF of Exponential 2
distribution has HF: HF of Exponential 2
distribution has mean: mean of Exponential 2
distribution has median: median of Exponential 2
distribution has mode: mode of Exponential 2
distribution has variance: variance of Exponential 2
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: scale of Exponential-2

Negative Binomial 4
type: probability distribution
has name: Negative Binomial 4
has code name: NegativeBinomial4
distribution has PMF: PMF of Negative Binomial 4
distribution has CDF: CDF of Negative Binomial 4
distribution has mean: mean of Negative Binomial 4
distribution has mode: mode of Negative Binomial 4
distribution has variance: variance of Negative Binomial 4
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\} \text{ -- number of successes}
distribution has parameter: number of failures of Negative-Binomial-4
distribution has parameter: success probability of Negative-Binomial-4

relationship between Gamma 1 and Exponential 1 whereby k=1, \\theta=1/\\lambda
type: Reparameterisation
distribution relationship from: Gamma 1
distribution relationship to: Exponential 1
distribution relationship definition in Latex: k=1, \theta=1/\lambda
distribution relationship pair expression Latex: Gamma1(k,\theta) \rightarrow Exponential1(\lambda)
distribution relationship reference: \cite{forbes2011statistical}

variance of Zeta 1
type: mathematical object
has name: variance of Zeta 1
has Latex expression: \frac{\zeta(\alpha)\zeta(\alpha-2) - \zeta(\alpha-1)^2}{\zeta(\alpha)^2}

scaling parameter of Scaled-Inverse-Chi-Square
type: mathematical object
has name: scaling parameter of Scaled-Inverse-Chi-Square
has code name: ScaledInverseChiSquare1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma \in R^+
has short name: scaling parameter
has short code name: scale

scale of Birnbaum-Saunders-1
type: mathematical object
has name: scale of Birnbaum-Saunders-1
has code name: BirnbaumSaunders1.scale
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: scale
has short code name: scale

HF of Two-Sided Power 1
type: hazard function
has name: HF of Two-Sided Power 1
has Latex expression: \begin{cases} -\frac{n(x-a)^{n-1} (m-a)^{1-n}}{a-b+(m-a)(x-a)^n (m-a)^{-n}} &\text{ for } a < x < m \\ \frac{n}{b-x} &\text{ for } m \leq x < b \end{cases}
has R code expression: HF1 = function(x,a,b,m,n) { -(n*(x-a)^(n-1)*(m-a)^(1-n))/(a-b+(m-a)*(x-a)^n*(m-a)^(-n)) } HF2 = function(x,a,b,m,n) { n/(b-x) }

relationship between Negative Binomial 1 and Poisson 1 whereby \\mu = np, n \\rightarrow \\infty
type: Limiting
distribution relationship from: Negative Binomial 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \mu = np, n \rightarrow \infty
distribution relationship pair expression Latex: NegativeBinomial1(r,p) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalPoisson.pdf}

relationship between Log-Normal 7 and Log-Normal 3 whereby m = \\mu_N/\\sqrt{1+\\sigma_N^2/\\mu_N^2}, \\sigma = \\sqrt{\\log1+\\sigma_N^2/\\mu_N^2}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 3
distribution relationship definition in Latex: m = \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}, \sigma = \sqrt{\log(1+\sigma_N^2/\mu_N^2)}
distribution relationship pair expression Latex: LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal3(m,\sigma)
distribution relationship reference: ProbOnto spec

relationship between Log-Normal 4 and Log-Normal 3 whereby m = m, \\sigma = \\sqrt{\\logcv^2+1}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 3
distribution relationship definition in Latex: m = m, \sigma = \sqrt{\log(cv^2+1)}
distribution relationship pair expression Latex: LogNormal4(m,cv) \rightarrow LogNormal3(m,\sigma)
distribution relationship reference: ProbOnto spec

probability of Inverse-Binomial-1
type: mathematical object
has name: probability of Inverse-Binomial-1
has code name: InverseBinomial1.probability
has Latex symbol: p
parameter has definition expressed in plain text: 1/2 < p < 1
has short name: probability
has short code name: probability

variance of Negative Binomial 2
type: mathematical object
has name: variance of Negative Binomial 2
has Latex expression: \lambda (1 + \tau \lambda)

variance of F 1
type: mathematical object
has name: variance of F 1
has Latex expression: \frac{2n_2^2(n_1+n_2-2)}{n_1(n_2-2)^2(n_2-4)}, n_2>4

upper bound of Arcsine-2
type: mathematical object
has name: upper bound of Arcsine-2
has code name: Arcsine2.upperBound
has Latex symbol: b
parameter has definition expressed in plain text: b \in R, b > a
has short name: upper bound
has short code name: upperBound

Fisk
type: probability distribution
label: Log-Logistic 2
has name: Log-Logistic 2
has code name: LogLogistic2
distribution has PDF: PDF of Log-Logistic 2
distribution has CDF: CDF of Log-Logistic 2
distribution has SF: SF of Log-Logistic 2
distribution has HF: HF of Log-Logistic 2
distribution has mean: mean of Log-Logistic 2
distribution has median: median of Log-Logistic 2
distribution has variance: variance of Log-Logistic 2
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Log-Logistic-2
distribution has parameter: scale of Log-Logistic-2

scale of Log-Logistic-1
type: mathematical object
has name: scale of Log-Logistic-1
has code name: LogLogistic1.scale
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: scale
has short code name: scale

PMF of Geometric 1
type: probability mass function
has name: PMF of Geometric 1
has Latex expression: (1 - p)^k\,p
has R code expression: p*(1-p)^k

log Poisson intensity of Poisson-2
type: mathematical object
has name: log Poisson intensity of Poisson-2
has code name: Poisson2.logRate
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R
has short name: log Poisson intensity
has short code name: logRate

SF of Half Cauchy 1
type: survival function
has name: SF of Half Cauchy 1
has Latex expression: 1 - \frac2\pi \arctan(x)
has R code expression: 1 - 2/pi * atan(x)

mean of Inverse-Wishart 1
type: mathematical object
has name: mean of Inverse-Wishart 1
has Latex expression: \frac{\Psi}{\nu - p - 1} \text{ for }\nu > p + 1

relationship between Negative Binomial 5 and Negative Binomial 3 whereby \\mu = \\alpha / \\beta, \\phi = \\alpha
type: Reparameterisation
distribution relationship from: Negative Binomial 5
distribution relationship to: Negative Binomial 3
distribution relationship definition in Latex: \mu = \alpha / \beta, \phi = \alpha
distribution relationship pair expression Latex: NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial3(\mu, \phi)
distribution relationship reference: ProbOnto spec

mean of Zeta 1
type: mathematical object
has name: mean of Zeta 1
has Latex expression: \zeta(\alpha-1)/\zeta(\alpha)

degrees of freedom of Scaled-Inverse-Chi-Square
type: mathematical object
has name: degrees of freedom of Scaled-Inverse-Chi-Square
has code name: ScaledInverseChiSquare1.degreesOfFreedom
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu \in R^+
has short name: degrees of freedom
has short code name: degreesOfFreedom

PMF of Negative Binomial 4
type: probability mass function
has name: PMF of Negative Binomial 4
has Latex expression: \binom {k+r-1}k (1-p)^r p^k
has R code expression: choose(k+r-1,k) * (1-p)^r * p^k

shape of Birnbaum-Saunders-1
type: mathematical object
has name: shape of Birnbaum-Saunders-1
has code name: BirnbaumSaunders1.shape
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma > 0
has short name: shape
has short code name: shape

CDF of Two-Sided Power 1
type: cumulative density function
has name: CDF of Two-Sided Power 1
has Latex expression: \begin{cases} \frac{(x-a)^n (m-a)^{1-n}}{b-a} \text{ for } a < x < m \\ -\frac{a+b(b-x)^n (b-m)^{-n} - b-m(b-x)^n (b-m)^{-n}}{b-a} \text{ for } m \leq x < b \end{cases}
has R code expression: CDF1 = function(x,a,b,m,n) { ((x-a)^n*(m-a)^(1-n))/(b-a) } CDF2 = function(x,a,b,m,n) { -(a+b*(b-x)^n*(b-m)^(-n)-b-m*(b-x)^n*(b-m)^(-n))/(b-a) }

Log-Uniform 1
type: probability distribution
has name: Log-Uniform 1
has code name: LogUniform1
distribution has PDF: PDF of Log-Uniform 1
distribution has CDF: CDF of Log-Uniform 1
distribution has mean: mean of Log-Uniform 1
distribution has variance: variance of Log-Uniform 1
has variate symbol in Latex: x
has Latex expression of support: x \in (min,max)
distribution has parameter: maximum of Log-Uniform-1
distribution has parameter: minimum of Log-Uniform-1

PDF of Gamma 2
type: probability density function
has name: PDF of Gamma 2
has Latex expression: \frac{\mu^r x^{r-1} e^{-\mu x}}{\Gamma(r)}
has R code expression: (mu^r * x^(r-1) * exp(-mu*x)) / gamma(r)

relationship between Gamma 1 and Normal 1 whereby \\mu = k \\theta, \\sigma^2 = k^2 \\theta, \\theta \\rightarrow \\infty
type: Limiting
distribution relationship from: Gamma 1
distribution relationship to: Normal
distribution relationship definition in Latex: \mu = k \theta, \sigma^2 = k^2 \theta, \theta \rightarrow \infty
distribution relationship pair expression Latex: Gamma1(k,theta) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaNormal1.pdf}

relationship between Generalized Poisson 3 and Poisson 1 whereby \\alpha = 0, \\lambda=\\mu
type: Special case
distribution relationship from: Generalized Poisson 3
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \alpha = 0, \lambda=\mu
distribution relationship pair expression Latex: GeneralizedPoisson3(\mu,\alpha) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{hilbe2011negative} \\ \cite{famoye2006zero}

relationship between Log-Normal 4 and Log-Normal 7 whereby \\mu_N = m \\sqrt{cv^2 + 1}, \\sigma_N = m \\;cv\\,\\sqrt{cv^2 + 1}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu_N = m \sqrt{cv^2 + 1}, \sigma_N = m \;cv\,\sqrt{cv^2 + 1}
distribution relationship pair expression Latex: LogNormal4(m,cv) \rightarrow LogNormal7(\mu_N,\sigma_N)
distribution relationship reference: ProbOnto spec

PDF of Normal-inverse-gamma 1
type: probability density function
has name: PDF of Normal-inverse-gamma 1
has Latex expression: \frac {\sqrt{\lambda}} {\sigma\sqrt{2\pi} } \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} e^{ -\frac { 2\beta + \lambda(x - \mu)^2} {2\sigma^2} }
has R code expression: sqrt(lambda)/(sigma*sqrt(2*pi)) * beta^alpha/gamma(alpha) * (1/sigma^2)^(alpha + 1) * exp(- (2*beta+lambda*(x-mu)^2)/(2*sigma^2))

mode of F 1
type: mathematical object
has name: mode of F 1
has Latex expression: \frac{n_2(n_1-2)}{n_1(n_2+2)}, n_1>2

PDF of Log-Logistic 2
type: probability density function
has name: PDF of Log-Logistic 2
has Latex expression: \frac{\lambda \kappa(\lambda x)^{\kappa-1}}{(1+(\lambda x)^\kappa)^2}
has R code expression: (lambda*kappa*(lambda*x)^(kappa-1)) / (1+(lambda*x)^kappa)^2

Noncentral Beta 1
type: probability distribution
has name: Noncentral Beta 1
has code name: NoncentralBeta1
distribution has PDF: PDF of Noncentral Beta 1
distribution has CDF: CDF of Noncentral Beta 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,1]
distribution has parameter: shape parameter of Noncentral-Beta-1
distribution has parameter: noncentrality parameter of Noncentral-Beta-1

CDF of Gamma 2
type: cumulative density function
has name: CDF of Gamma 2
has Latex expression: \frac{1}{\Gamma(r)} \gamma\left(r, \mu x\right)
has R code expression: 1/gamma(r) * Igamma(r,mu*x,lower=T)

Inverse-Gamma 1
type: probability distribution
has name: Inverse-Gamma 1
has code name: InverseGamma1
distribution has PDF: PDF of Inverse-Gamma 1
distribution has CDF: CDF of Inverse-Gamma 1
distribution has mean: mean of Inverse-Gamma 1
distribution has mode: mode of Inverse-Gamma 1
distribution has variance: variance of Inverse-Gamma 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Inverse-Gamma-1
distribution has parameter: scale of Inverse-Gamma-1

noncentrality parameter of Rice-1
type: mathematical object
has name: noncentrality parameter of Rice-1
has code name: Rice1.noncentrality
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu \geq 0
has short name: noncentrality parameter
has short code name: noncentrality

Transformation
type: Class
subclass of: relationship between distributions

relationship between Exponential 2 and Exponential 1 whereby \\lambda=1/\\beta
type: Reparameterisation
distribution relationship from: Negative exponential
distribution relationship to: Exponential 1
distribution relationship definition in Latex: \lambda=1/\beta
distribution relationship pair expression Latex: Exponential2(\beta) \rightarrow Exponential1(\lambda)
distribution relationship reference: ProbOnto spec

Trapezoidal 1
type: probability distribution
label: Piecewise Random
has name: Trapezoidal 1
has code name: Trapezoidal1
distribution has PDF: PDF of Trapezoidal 1
distribution has CDF: CDF of Trapezoidal 1
distribution has mean: mean of Trapezoidal 1
distribution has variance: variance of Trapezoidal 1
has variate symbol in Latex: x
has Latex expression of support: x \in [a,d]
distribution has parameter: Upper bound of Trapezoidal-1
distribution has parameter: Lower bound of Trapezoidal-1
distribution has parameter: Level end of Trapezoidal-1
distribution has parameter: Level start of Trapezoidal-1

relationship between Gamma 1 and Inverse-Gamma 1 whereby \\text{If } X \\sim Gamma1\\alpha,\\beta \\text{ then } X^{-1} \\sim InverseGamma1\\alpha,\\beta^{-1}
type: Transformation
distribution relationship from: Gamma 1
distribution relationship to: Inverse-Gamma 1
distribution relationship definition in Latex: \text{If } X \sim Gamma1(\alpha,\beta) \text{ then } X^{-1} \sim InverseGamma1(\alpha,\beta^{-1})
distribution relationship pair expression Latex: Gamma1(\alpha,\beta) \rightarrow InverseGamma1(\alpha,\beta)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Inverse-gamma_distribution}

mode of Inverse-Wishart 1
type: mathematical object
has name: mode of Inverse-Wishart 1
has Latex expression: \frac{\Psi}{\nu + p + 1}

PDF of Exponentially modified Gaussian 1
type: probability density function
has name: PDF of Exponentially modified Gaussian 1
has Latex expression: \frac{\lambda}{2} e^{\frac{\lambda}{2} (2 \mu + \lambda \sigma^2 - 2 x)} \mbox{erfc} (\frac{\mu + \lambda \sigma^2 - x}{ \sqrt{2} \sigma})
has R code expression: lambda/2*exp(lambda/2*(2*mu+lambda*sigma^2-2*x))*erfc((mu+lambda*sigma^2-x)/(sqrt(2)*sigma)) \\ erfc <- function(x) 2 * pnorm(x * sqrt(2), lower = FALSE)

variance of Poisson 2
type: mathematical object
has name: variance of Poisson 2
has Latex expression: \exp(\alpha)

Dagum 1
type: probability distribution
has name: Dagum 1
has code name: Dagum1
distribution has PDF: PDF of Dagum 1
distribution has CDF: CDF of Dagum 1
distribution has mean: mean of Dagum 1
distribution has median: median of Dagum 1
distribution has mode: mode of Dagum 1
distribution has variance: variance of Dagum 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: scale parameter of Dagum-1
distribution has parameter: shape parameter of Dagum-1

shape of Log-Logistic-1
type: mathematical object
has name: shape of Log-Logistic-1
has code name: LogLogistic1.shape
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape
has short code name: shape

scale of Lomax-1
type: mathematical object
has name: scale of Lomax-1
has code name: Lomax1.scale
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0
has short name: scale
has short code name: scale

Negative Binomial 6
type: probability distribution
has name: Negative Binomial 6
has code name: NegativeBinomial6
distribution has PMF: PMF of Negative Binomial 6
distribution has mean: mean of Negative Binomial 6
distribution has variance: variance of Negative Binomial 6
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: log mean of Negative-Binomial-6
distribution has parameter: index parameter of Negative-Binomial-6

distribution has SF
type: ObjectProperty

relationship between Frechet 2 and Frechet 1 whereby m=0
type: Special case
distribution relationship from: Inverse Weibull
distribution relationship to: Frechet 1
distribution relationship definition in Latex: m=0
distribution relationship pair expression Latex: Frechet2(\alpha,\sigma,m) \rightarrow Frechet1(\alpha,\sigma)
distribution relationship reference: \url{http://www.mathwave.com/help/easyfit/html/analyses/distributions/frechet.html}

Double Exponential 1
type: probability distribution
label: Laplace 1
has name: Laplace 1
has code name: Laplace1
distribution has PDF: PDF of Laplace 1
distribution has CDF: CDF of Laplace 1
distribution has mean: mean of Laplace 1
distribution has median: median of Laplace 1
distribution has mode: mode of Laplace 1
distribution has variance: variance of Laplace 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location of Laplace-1
distribution has parameter: scale of Laplace-1

HF of Exponential 1
type: hazard function
has name: HF of Exponential 1
has Latex expression: \lambda
has R code expression: lambda

scale of Erlang-1
type: mathematical object
has name: scale of Erlang-1
has code name: Erlang1.scale
has Latex symbol: b
parameter has definition expressed in plain text: b>0
has short name: scale
has short code name: scale

PROB k0000959
type: Transformation
label: relationship between Normal 1 and Logit Normal 1 whereby \\text{ If } Y \\text{ is a random variable with a normal distribution, and } P \\text{ is the logistic function, then } \\\\ X = PY \\text{ has a logit-normal distribution}
distribution relationship from: Normal
distribution relationship to: Logit Normal 1
distribution relationship definition in Latex: \text{ If } Y \text{ is a random variable with a normal distribution, and } P \text{ is the logistic function, then } \\ X = P(Y) \text{ has a logit-normal distribution}
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow LogitNormal1(\mu,\sigma)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Logit-normal_distribution}

Normal-inverse-gamma 1
type: probability distribution
label: Gaussian- inverse-gamma
has name: Normal-inverse-gamma 1
has code name: NormalInverseGamma1
distribution has PDF: PDF of Normal-inverse-gamma 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty), \sigma^2 \in (0,+\infty)
distribution has parameter: location of Normal-inverse-gamma-1
distribution has parameter: scale of Normal-inverse-gamma-1
distribution has parameter: shape of Normal-inverse-gamma-1
distribution has parameter: lambda of Normal-inverse-gamma-1

variance of Normal 3
type: mathematical object
has name: variance of Normal 3
has Latex expression: 1/\tau

PDF of Inverse-Gamma 1
type: probability density function
has name: PDF of Inverse-Gamma 1
has Latex expression: \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)
has R code expression: beta^alpha/gamma(alpha) * x^(-alpha-1) * exp(-beta/x)

scale parameter of Rice-1
type: mathematical object
has name: scale parameter of Rice-1
has code name: Rice1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: scale parameter
has short code name: scale

relationship between Log-Normal 7 and Log-Normal 2 whereby \\mu = \\log\\Big \\mu_N/\\sqrt{1+\\sigma_N^2/\\mu_N^2} \\Big, v = \\log1+\\sigma_N^2/\\mu_N^2
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 2
distribution relationship definition in Latex: \mu = \log\Big( \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2} \Big), v = \log(1+\sigma_N^2/\mu_N^2)
distribution relationship pair expression Latex: LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal2(\mu,v)
distribution relationship reference: ProbOnto spec

mean of Zipf 1
type: mathematical object
has name: mean of Zipf 1
has Latex expression: H_{n,\alpha-1}/ H_{n,\alpha}

relationship between Generalized Gamma 3 and Generalized Gamma 1 whereby a=1/\\mu, d=\\beta r, p=\\beta
type: Reparameterisation
distribution relationship from: Generalized Gamma 3
distribution relationship to: Generalized Gamma 1
distribution relationship definition in Latex: a=1/\mu, d=\beta r, p=\beta
distribution relationship pair expression Latex: GeneralizedGamma3(r,\mu,\beta) \rightarrow GeneralizedGamma1(a,d,p)
distribution relationship reference: ProbOnto spec

PDF of Trapezoidal 1
type: probability density function
has name: PDF of Trapezoidal 1
has Latex expression: \left\{ \begin{array}{rcl} \dfrac{2}{d+c-a-b}\dfrac{x-a}{b-a} & a\leq x < b & \\ \dfrac{2}{d+c-a-b} & b\leq x < c \\ \dfrac{2}{d+c-a-b}\dfrac{d-x}{d-c} & c\leq x \leq d \end{array}\right. \nonumber \\
has R code expression: PMF1 = 2/(d+c-a-b)*(x-a)/(b-a) PMF2 = 2/(d+c-a-b) PMF3 = 2/(d+c-a-b)*(d-x)/(d-c)

Transformation, Limiting
type: Class
subclass of: relationship between distributions

shape of Zeta-1
type: mathematical object
has name: shape of Zeta-1
has code name: Zeta1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R, \alpha > 0
has short name: shape
has short code name: shape

relationship between Hypergeometric 1 and Binomial 1 whereby p = K/N, n=n, N \\rightarrow \\infty
type: Limiting
distribution relationship from: Hypergeometric 1
distribution relationship to: Binomial 1
distribution relationship definition in Latex: p = K/N, n=n, N \rightarrow \infty
distribution relationship pair expression Latex: Hypergeometric1(N,K,n) \rightarrow Binomial1(n,p)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/HypergeometricBinomial.pdf}

ExGaussian
type: probability distribution
label: Exponentially modified Gaussian 1
has name: Exponentially modified Gaussian 1
has code name: ExponentiallyModifiedGaussian1
distribution has PDF: PDF of Exponentially modified Gaussian 1
distribution has CDF: CDF of Exponentially modified Gaussian 1
distribution has mean: mean of Exponentially modified Gaussian 1
distribution has variance: variance of Exponentially modified Gaussian 1
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: standard deviation of Exponentially-modified-Gaussian-1
distribution has parameter: mean of Exponentially-modified-Gaussian-1
distribution has parameter: rate or inverse scale of Exponentially-modified-Gaussian-1

scale matrix of Inverse-Wishart-1
type: mathematical object
has name: scale matrix of Inverse-Wishart-1
has code name: InverseWishart1.scaleMatrix
has Latex symbol: \Psi
parameter has definition expressed in plain text: \Psi > 0, \text{positive-definite matrix}
has short name: scale matrix
has short code name: scaleMatrix

Pareto Type II
type: probability distribution
has name: Pareto Type II
has code name: ParetoTypeII1
distribution has PDF: PDF of Pareto Type II
distribution has CDF: CDF of Pareto Type II
has variate symbol in Latex: x
has Latex expression of support: x \in [\mu, +\infty)
distribution has parameter: location of Pareto-Type-II
distribution has parameter: tail index of Pareto-Type-II
distribution has parameter: scale of Pareto-Type-II

variance of Lomax 1
type: mathematical object
has name: variance of Lomax 1
has Latex expression: {{\lambda^2 \alpha} \over {(\alpha-1)^2(\alpha-2)}} \text{ for } \alpha > 2;\\ \infty \text{ for } 1 < \alpha \le 2; \\ \text{Otherwise undefined}

mean of Poisson 2
type: mathematical object
has name: mean of Poisson 2
has Latex expression: \exp(\alpha)

distribution has HF
type: ObjectProperty

mean of Inverse-Gaussian-1
type: mathematical object
has name: mean of Inverse-Gaussian-1
has code name: InverseGaussian1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu > 0
has short name: mean
has short code name: mean

CDF of Exponential 1
type: cumulative density function
has name: CDF of Exponential 1
has Latex expression: 1 - \exp(-\lambda x)
has R code expression: 1 - exp(-lambda*x)

shape of Erlang-1
type: mathematical object
has name: shape of Erlang-1
has code name: Erlang1.shape
has Latex symbol: c
parameter has definition expressed in plain text: c>0
has short name: shape
has short code name: shape

mean of Normal-3
type: mathematical object
has name: mean of Normal-3
has code name: Normal3.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: mean

PDF of Laplace 1
type: probability density function
has name: PDF of Laplace 1
has Latex expression: \frac{1}{2\,b} \exp \left(-\frac{|x-\mu|}b \right)
has R code expression: 1/(2*b) * exp(- abs(x-mu)/b )

CDF of Zipf 1
type: cumulative density function
has name: CDF of Zipf 1
has Latex expression: H_{x,\alpha}/H_{n,\alpha}, \text{ with } x=1,2,...,n\\ Hfunc = function(n,alpha) { Hsum=0; for(i in 1:n) { Hsum = Hsum + (1/i)^alpha } return(Hsum) }
has R code expression: CDF = function(x,n,alpha) { H=array(0,length(x)); for(j in 1:length(x)) { H[j] = Hfunc(x[j],alpha) / Hfunc(n,alpha) } cat(H) return(H) }

scale of Logistic-1
type: mathematical object
has name: scale of Logistic-1
has code name: Logistic1.scale
has Latex symbol: s
parameter has definition expressed in plain text: s > 0, s \in R
has short name: scale
has short code name: scale

relationship between Log-Normal 3 and Log-Normal 7 whereby \\mu_N = m\\;\\exp\\sigma^2/2, \\sigma_N = m \\;\\exp\\sigma^2/2\\sqrt{\\exp\\sigma^2-1}
type: Reparameterisation
distribution relationship from: Log-Normal 3
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu_N = m\;\exp(\sigma^2/2), \sigma_N = m \;\exp(\sigma^2/2)\sqrt{\exp(\sigma^2)-1}
distribution relationship pair expression Latex: LogNormal3(m,\sigma) \rightarrow LogNormal7(\mu_N,\sigma_N)
distribution relationship reference: ProbOnto spec

CDF of Inverse-Gamma 1
type: cumulative density function
has name: CDF of Inverse-Gamma 1
has Latex expression: \frac{\Gamma(\alpha, \beta/x)}{\Gamma(\alpha)}
has R code expression: Igamma(alpha, beta/x, lower=F) / gamma(alpha)

degrees of freedom of Inverse-Wishart-1
type: mathematical object
has name: degrees of freedom of Inverse-Wishart-1
has code name: InverseWishart1.degreesOfFreedom
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu > p-1, \nu \in R
has short name: degrees of freedom
has short code name: degreesOfFreedom

Limiting, Special case
type: Class
subclass of: relationship between distributions

relationship between Log-Normal 7 and Log-Normal 1 whereby \\mu = \\log\\Big\\mu_N/\\sqrt{1+\\sigma_N^2/\\mu_N^2}\\Big, \\sigma = \\sqrt{\\log1+\\sigma_N^2/\\mu_N^2}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \mu = \log\Big(\mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}\Big), \sigma = \sqrt{\log(1+\sigma_N^2/\mu_N^2)}
distribution relationship pair expression Latex: LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

mean of Exponentially modified Gaussian 1
type: mathematical object
has name: mean of Exponentially modified Gaussian 1
has Latex expression: \mu + 1/\lambda

relationship between Logistic 1 and Logistic 2 whereby \\tau=1/s
type: Reparameterisation
distribution relationship from: Logistic 1
distribution relationship to: Logistic 2
distribution relationship definition in Latex: \tau=1/s
distribution relationship pair expression Latex: Logistic1(\mu,s) \rightarrow Logistic2(\mu,\tau)
distribution relationship reference: ProbOnto spec

PDF of Pareto Type II
type: probability density function
has name: PDF of Pareto Type II
has Latex expression: {\alpha \over \lambda} \left[{1+ {x-\mu \over \lambda}}\right]^{-(\alpha+1)}
has R code expression: alpha/lambda* (1+ (x-mu)/lambda)^(-(alpha+1))

location of Von-Mises-1
type: mathematical object
has name: location of Von-Mises-1
has code name: VonMises1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

CDF of Generalized Poisson 3
type: cumulative density function
has name: CDF of Generalized Poisson 3
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

location of Logistic-1
type: mathematical object
has name: location of Logistic-1
has code name: Logistic1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

PDF of F 1
type: probability density function
has name: PDF of F 1
has Latex expression: \frac{\Gamma(\frac{n_1 + n_2}{2}) (\frac{n_1}{n_2})^{n_1/2} x^{n_1/2-1}}{\Gamma(\frac{n_1}{2})\Gamma(\frac{n_2}{2})\big[\frac{n_1}{n_2}x+1\big]^{(n_1+n_2)/2}}
has R code expression: gamma((n1 + n2)/2)*(n1/n2)^(n1/2)*x^(n1/2-1)/(gamma(n1/2)*gamma(n2/2)*(n1/n2*x+1)^((n1+n2)/2))

mean of Inverse-Gamma 1
type: mathematical object
has name: mean of Inverse-Gamma 1
has Latex expression: \frac{\beta}{\alpha-1} \text{ for } \alpha > 1

CDF of Laplace 1
type: cumulative density function
has name: CDF of Laplace 1
has Latex expression: \begin{cases} \frac12 \exp \left( \frac{x-\mu}{b} \right) & \mbox{if }x < \mu \\ 1-\frac12 \exp \left( -\frac{x-\mu}{b} \right) & \mbox{if }x \geq \mu \end{cases}
has R code expression: 1/2 * exp( (x-mu)/b ) # for x < mu 1- 1/2 * exp( -(x-mu)/b ) # x >= mu

relationship between Generalized Gamma 2 and Gamma 1 whereby k=1, a=0 \\text{ and renaming parameters: } c=k, b=\\theta
type: Transformation
distribution relationship from: Generalized Gamma 2
distribution relationship to: Gamma 1
distribution relationship definition in Latex: k=1, a=0 \text{ and renaming parameters: } c=k, b=\theta
distribution relationship pair expression Latex: GeneralizedGamma2(a,b,c,k) \rightarrow Gamma1(k,\theta)
distribution relationship reference: \cite{forbes2011statistical}

relationship between Log-Normal 5 and Log-Normal 4 whereby m = \\exp\\mu, cv = \\sqrt{\\exp1/\\tau - 1}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m = \exp(\mu), cv = \sqrt{\exp(1/\tau) - 1}
distribution relationship pair expression Latex: LogNormal5(\mu,\tau) \rightarrow LogNormal4(m,cv)
distribution relationship reference: ProbOnto spec

precision of Normal-3
type: mathematical object
has name: precision of Normal-3
has code name: Normal3.precision
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau>0
has short name: precision
has short code name: precision

mean of Noncentral chi-squared 1
type: mathematical object
has name: mean of Noncentral chi-squared 1
has Latex expression: \delta + n

CDF of Exponentially modified Gaussian 1
type: cumulative density function
has name: CDF of Exponentially modified Gaussian 1
has Latex expression: \Phi(u, 0, v) - e^{-u + v^2/2+ \log(\Phi(u, v^2, v))} \text{ with } u = \lambda(x - \mu) \text{ and } v = \lambda \sigma
has R code expression: CDF = function(x,mu,sigma,lambda) { u = lambda * (x - mu); v = lambda * sigma; CDF_Normal1(u, 0, v) - exp(-u + v^2/2 + log(CDF_Normal1(u, v^2, v)))}; CDF_Normal1 = function(x,mu,sigma) { 1/2 * (1 + erf((x-mu)/(sigma*sqrt(2)))) }

Normal
type: probability distribution
label: Gaussian
label: Normal 1
has name: Normal 1
has code name: Normal1
distribution has PDF: PDF of Normal 1
distribution has CDF: CDF of Normal 1
distribution has mean: mean of Normal 1
distribution has median: median of Normal 1
distribution has mode: mode of Normal 1
distribution has variance: variance of Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: standard deviation of Normal-1
distribution has parameter: mean of Normal-1

Special case, Limiting
type: Class
subclass of: relationship between distributions

relationship between Log-Normal 2 and Log-Normal 7 whereby \\mu_N = \\exp\\mu+v/2, \\sigma_N = \\exp\\mu+v/2\\sqrt{\\expv-1}
type: Reparameterisation
distribution relationship from: Log-Normal 2
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu_N = \exp(\mu+v/2), \sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}
distribution relationship pair expression Latex: LogNormal2(\mu,v) \rightarrow LogNormal7(\mu_N,\sigma_N)
distribution relationship reference: ProbOnto spec

relationship between Erlang 1 and Exponential 2 whereby c=1, b=\\beta
type: Special case
distribution relationship from: Erlang 1
distribution relationship to: Negative exponential
distribution relationship definition in Latex: c=1, b=\beta
distribution relationship pair expression Latex: Erlang1(b,c) \rightarrow Exponential2(\beta)
distribution relationship reference: \cite{forbes2011statistical}

PMF of Negative Binomial 6
type: probability mass function
has name: PMF of Negative Binomial 6
has Latex expression: \binom {k+\phi-1}k \Big(\frac{\exp(\eta)}{\exp(\eta) + \phi} \Big)^{k} \Big(\frac{\phi}{\phi + \exp(\eta)} \Big)^{\phi}
has R code expression: choose(k+phi-1,k) * (exp(eta) / (exp(eta) + phi))^k * (phi / (phi + exp(eta)))^phi

CDF of Pareto Type II
type: cumulative density function
has name: CDF of Pareto Type II
has Latex expression: 1- \left[{1+ {x-\mu \over \lambda}}\right]^{-\alpha}
has R code expression: 1- (1+ (x-mu)/lambda)^(-alpha)

mode of Arcsine 2
type: mathematical object
has name: mode of Arcsine 2
has Latex expression: x \in \{a,b\}

variance of Logistic 1
type: mathematical object
has name: variance of Logistic 1
has Latex expression: \frac{s^2 \pi^2}{3}

variance of Skew Normal
type: mathematical object
has name: variance of Skew Normal
has Latex expression: \sigma^2\left(1 - \frac{2\delta^2}{\pi}\right) \text{ where } \delta = \frac{\alpha}{\sqrt{1+\alpha^2}}

relationship between Standard Uniform 1 and Pareto Type I whereby x_m X^{-1/\\alpha}
type: Transformation
distribution relationship from: Standard Uniform 1
distribution relationship to: Pareto Type I
distribution relationship definition in Latex: x_m X^{-1/\alpha}
distribution relationship pair expression Latex: StandardUniform1(0,1) \rightarrow ParetoTypeI1(x_m,\alpha)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformPareto.pdf}

mode of Inverse-Gamma 1
type: mathematical object
has name: mode of Inverse-Gamma 1
has Latex expression: \frac{\beta}{\alpha + 1}

mean of Laplace 1
type: mathematical object
has name: mean of Laplace 1
has Latex expression: \mu

F 1
type: probability distribution
label: Fisher-Snedecor
has name: F 1
has code name: F1
distribution has PDF: PDF of F 1
distribution has CDF: CDF of F 1
distribution has mean: mean of F 1
distribution has mode: mode of F 1
distribution has variance: variance of F 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: degree of freedom of F-1

relationship between Zero-inflated Poisson 1 and Poisson 1 whereby \\pi=0
type: Special case
distribution relationship from: Zero-inflated Poisson 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \pi=0
distribution relationship pair expression Latex: ZeroInflatedPoisson1(\lambda,\pi) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{Troconiz:2009fv}

CDF of Frechet 1
type: cumulative density function
has name: CDF of Frechet 1
has Latex expression: \exp\Big(-\Big(\frac{\sigma}{x}\Big)^\alpha\Big)
has R code expression: exp(-(sigma/x)^alpha)

Chi-square
type: probability distribution
label: Chi-squared 1
label: Chi square
label: Chisquare
has name: Chi-squared 1
has code name: ChiSquared1
distribution has PDF: PDF of Chi-squared 1
distribution has CDF: CDF of Chi-squared 1
distribution has mean: mean of Chi-squared 1
distribution has median: median of Chi-squared 1
distribution has mode: mode of Chi-squared 1
distribution has variance: variance of Chi-squared 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: degrees of freedom of Chi-squared-1

PROB k0000821
type: Transformation
label: relationship between Conway-Maxwell-Poisson 1 and Binomial 1 whereby \\text{For } \\nu=\\infty \\text{ the distribution of the sum is binomial with parameters } \\\\ n \\text{ and } \\lambda/1+\\lambda
distribution relationship from: Conway-Maxwell-Poisson 1
distribution relationship to: Binomial 1
distribution relationship definition in Latex: \text{For } \nu=\infty \text{ the distribution of the sum is binomial with parameters } \\ n \text{ and } \lambda/(1+\lambda)
distribution relationship pair expression Latex: ConwayMaxwellPoisson1(\lambda,\nu) \rightarrow Binomial1(p)
distribution relationship reference: \cite{shmueli2005useful}

shape of Frechet-2
type: mathematical object
has name: shape of Frechet-2
has code name: Frechet2.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in (0, \infty)
has short name: shape
has short code name: shape

CDF of Skew Normal
type: cumulative density function
has name: CDF of Skew Normal
has Latex expression: \Phi\Big(\frac{x-\mu}{\sigma}\Big) - 2 T\Big(\frac{x-\mu}{\sigma},\alpha\Big)
has R code expression: Phi((x-mu)/sigma) - 2*T.Owen((x-mu)/sigma,alpha)

PROB k0001416
type: Limiting
label: relationship between Zipf 1 and Zeta 1 whereby \\text{The limiting distribution of a } Zipf1\\alpha, n \\text{ random variable as }\\\\ n\\rightarrow \\infty \\text{ is the } Zeta1\\alpha \\text{ distribution.}
distribution relationship from: Zipf 1
distribution relationship to: Zeta 1
distribution relationship definition in Latex: \text{The limiting distribution of a } Zipf1(\alpha, n) \text{ random variable as }\\ n\rightarrow \infty \text{ is the } Zeta1(\alpha) \text{ distribution.}
distribution relationship pair expression Latex: Zipf1(\alpha,n) \rightarrow Zeta1(\alpha)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ZipfZeta.pdf}

mean of Arcsine 2
type: mathematical object
has name: mean of Arcsine 2
has Latex expression: (a+b)/2

shape parameter of Beta-Negative-Binomial1
type: mathematical object
has name: shape parameter of Beta-Negative-Binomial1
has code name: BetaNegativeBinomial1.shape2
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape parameter
has short code name: shape2

variance of Logistic 2
type: mathematical object
has name: variance of Logistic 2
has Latex expression: \frac{\pi^2}{3\tau^2 }

relationship between Student's t-distribution 3 and Student's t-distribution 1 whereby \\mu=0, \\sigma=1
type: Special case
distribution relationship from: non-standardized Student's t-distribution
distribution relationship to: t-distribution
distribution relationship definition in Latex: \mu=0, \sigma=1
distribution relationship pair expression Latex: StudentT3(\nu,\mu,\sigma) \rightarrow StudentT1(\nu)
distribution relationship reference: ProbOnto spec

has R code expression
type: DatatypeProperty

cumulative density function
type: Class
subclass of: mathematical function

Normal
type: probability distribution
label: Gaussian
label: Normal 2
has name: Normal 2
has code name: Normal2
distribution has PDF: PDF of Normal 2
distribution has CDF: CDF of Normal 2
distribution has mean: mean of Normal 2
distribution has median: median of Normal 2
distribution has mode: mode of Normal 2
distribution has variance: variance of Normal 2
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: variance of Normal-2
distribution has parameter: mean of Normal-2

PROB k0000938
type: Transformation
label: relationship between Chi-squared 1 and F 1 whereby \\text{If } X_1 \\sim ChiSquared1n_1, X_2 \\sim ChiSquared1n_2 \\text{ are independent random variables }\\\\ \\Rightarrow \\frac{X_1/n_1}{X_2/n_2} \\sim F1n_1,n_2
distribution relationship from: Chi-square
distribution relationship to: F 1
distribution relationship definition in Latex: \text{If } X_1 \sim ChiSquared1(n_1), X_2 \sim ChiSquared1(n_2) \text{ are independent random variables }\\ \Rightarrow \frac{X_1/n_1}{X_2/n_2} \sim F1(n_1,n_2)
distribution relationship pair expression Latex: ChiSquared1(n) \rightarrow F1(n_1,n_2)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ChisquareF.pdf}

PDF of Triangular 1
type: probability density function
has name: PDF of Triangular 1
has Latex expression: \begin{cases} 2(x-a) / [(b-a)(c-a)] & \text{ for } a \leq x \leq c \\ 2(b-x) / [(b-a)(b-c)] & \text{ for } c \leq x \leq b \end{cases}
has R code expression: 2*(x-a) / ((b-a)*(c-a)) for a <= x <= c \\ 2*(b-x) / ((b-a)*(b-c)) for c <= x <= b

relationship between Negative Binomial 1 and Negative Binomial 3 whereby \\phi=r, \\mu=r1-p/p
type: Reparameterisation
distribution relationship from: Negative Binomial 1
distribution relationship to: Negative Binomial 3
distribution relationship definition in Latex: \phi=r, \mu=r(1-p)/p
distribution relationship pair expression Latex: NegativeBinomial1(r, p) \rightarrow NegativeBinomial3(\mu, \phi)
distribution relationship reference: ProbOnto spec

PDF of Student's t-distribution 1
type: probability density function
has name: PDF of Student's t-distribution 1
has Latex expression: \frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}
has R code expression: gamma((nu+1)/2)/(sqrt(nu*pi)*gamma(nu/2))*(1+x^2/nu)^(-(nu+1)/2)

number of successes of Beta-Negative-Binomial1
type: mathematical object
has name: number of successes of Beta-Negative-Binomial1
has code name: BetaNegativeBinomial1.numberOfSuccesses
has Latex symbol: n
parameter has definition expressed in plain text: n \in N
has short name: number of successes
has short code name: numberOfSuccesses

location of Standard-Two-Sided-Power-1
type: mathematical object
has name: location of Standard-Two-Sided-Power-1
has code name: StandardTwoSidedPower1.location
has Latex symbol: m
parameter has definition expressed in plain text: a < m < b
has short name: location
has short code name: location

distribution has variance
type: ObjectProperty

shape of Gamma-1
type: mathematical object
has name: shape of Gamma-1
has code name: Gamma1.shape
has Latex symbol: k
parameter has definition expressed in plain text: k > 0
has short name: shape
has short code name: shape

relationship between distributions
type: Class

number
type: Class
subclass of: mathematical object

PDF of Log-Logistic 1
type: probability density function
has name: PDF of Log-Logistic 1
has Latex expression: \frac{(\beta/\alpha)(x/\alpha)^{\beta-1}}{(1+(x/\alpha)^\beta)^2}
has R code expression: (beta/alpha)*(x/alpha)^(beta-1) / (1+(x/alpha)^beta)^2

Rayleigh 1
type: probability distribution
has name: Rayleigh 1
has code name: Rayleigh1
distribution has PDF: PDF of Rayleigh 1
distribution has CDF: CDF of Rayleigh 1
distribution has mean: mean of Rayleigh 1
distribution has median: median of Rayleigh 1
distribution has mode: mode of Rayleigh 1
distribution has variance: variance of Rayleigh 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: scale of Rayleigh-1

relationship between Log-Normal 1 and Log-Normal 7 whereby \\mu_N = \\exp\\Big\\mu + \\frac12 \\sigma^2\\Big, \\sigma_N = \\exp\\big\\mu + \\frac{1}{2}\\sigma^2\\big\\sqrt{\\exp\\sigma^2-1}
type: Reparameterisation
distribution relationship from: Log-Normal 1
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu_N = \exp\Big(\mu + \frac12 \sigma^2\Big), \sigma_N = \exp\big(\mu + \frac{1}{2}\sigma^2\big)\sqrt{\exp(\sigma^2)-1}
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow LogNormal7(\mu_N,\sigma_N)
distribution relationship reference: ProbOnto spec

PROB k0001417
type: Special case
label: relationship between Triangular 1 and Standard Triangular 1 whereby StandardTriangular1 \\text{ distribution is a special case of the } Triangular1a,b,c \\text{ distribution } \\\\ \\text{ when } a = -1, b = 1, c = 0.
distribution relationship from: Triangular 1
distribution relationship to: Standard Triangular 1
distribution relationship definition in Latex: StandardTriangular1 \text{ distribution is a special case of the } Triangular1(a,b,c) \text{ distribution } \\ \text{ when } a = -1, b = 1, c = 0.
distribution relationship pair expression Latex: Triangular1(a,b,c) \rightarrow StandardTriangular1
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TriangularStandardtriangular.pdf}

mean of Skew Normal
type: mathematical object
has name: mean of Skew Normal
has Latex expression: \mu + \sigma\delta\sqrt{\frac{2}{\pi}} \text{ where } \delta = \frac{\alpha}{\sqrt{1+\alpha^2}}

CDF of Arcsine 2
type: cumulative density function
has name: CDF of Arcsine 2
has Latex expression: \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right)
has R code expression: 2/pi*asin(sqrt((x-a)/(b-a)))

location of Logistic-2
type: mathematical object
has name: location of Logistic-2
has code name: Logistic2.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

scale of Laplace-1
type: mathematical object
has name: scale of Laplace-1
has code name: Laplace1.scale
has Latex symbol: b
parameter has definition expressed in plain text: b > 0, b \in R
has short name: scale
has short code name: scale

probability mass function
type: Class
subclass of: mathematical function

has abbreviation
type: DatatypeProperty

relationship between Negative Binomial 2 and Negative Binomial 1 whereby r=1/\\tau, p = 1/1+\\tau \\lambda
type: Reparameterisation
distribution relationship from: Negative Binomial 2
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: r=1/\tau, p = 1/(1+\tau \lambda)
distribution relationship pair expression Latex: NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial1(r, p)
distribution relationship reference: ProbOnto spec

PROB k0001436
type: Transformation
label: relationship between Normal 1 and Maxwell Boltzmann 1 whereby \\text{If } U_1, U_2, U_3 \\text{ are independent normal variables with mean} 0 \\text{ and standard deviation } \\sigma > 0 \\text{ then } X = \\sqrt{U_1^2 +U_3^2 +U_3^2} \\text{ has the Maxwell distribution with scale parameter } \\sigma
distribution relationship from: Normal
distribution relationship to: Maxwell Boltzmann 1
distribution relationship definition in Latex: \text{If } U_1, U_2, U_3 \text{ are independent normal variables with mean} 0 \text{ and standard deviation } \sigma > 0 \text{ then } X = \sqrt{U_1^2 +U_3^2 +U_3^2} \text{ has the Maxwell distribution with scale parameter } \sigma
distribution relationship pair expression Latex: Normal1(\mu, \sigma) \rightarrow MaxwellBoltzmann1(\sigma)
distribution relationship reference: \url{http://www.math.uah.edu/stat/special/Maxwell.html}

CDF of Student's t-distribution 1
type: cumulative density function
has name: CDF of Student's t-distribution 1
has Latex expression: \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right) \times \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2}; -\frac{x^2}{\nu} \right)} {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)}
has R code expression: 1/2+x*gamma((nu+1)/2)*hypergeo(1/2,(nu+1)/2,3/2,-x^2/nu)/( sqrt(pi*nu) *gamma(nu/2))

CDF of Triangular 1
type: cumulative density function
has name: CDF of Triangular 1
has Latex expression: \begin{cases} (x-a)^2 / [(b-a)(c-a)] & \text{ for } a \leq x \leq c \\ 1 - (b-x)^2 / [(b-a)(b-c)] & \text{ for } c \leq x \leq b \end{cases}
has R code expression: (x-a)^2 / ((b-a)*(c-a)) for a <= x <= c \\ 1 - (b-x)^2 / ((b-a)*(b-c)) for c <= x <= b

relationship between Student's t-distribution 2 and Student's t-distribution 1 whereby \\mu=0, \\tau=1
type: Reparameterisation
distribution relationship from: t-distribution
distribution relationship to: t-distribution
distribution relationship definition in Latex: \mu=0, \tau=1
distribution relationship pair expression Latex: StudentT2(\mu,\tau,k) \rightarrow StudentT1(\nu)

HF of Geometric 1
type: hazard function
has name: HF of Geometric 1
has Latex expression: p
has R code expression: p

distribution has mode
type: ObjectProperty

Log-Logistic 1
type: probability distribution
label: Fisk
has name: Log-Logistic 1
has code name: LogLogistic1
distribution has PDF: PDF of Log-Logistic 1
distribution has CDF: CDF of Log-Logistic 1
distribution has mean: mean of Log-Logistic 1
distribution has median: median of Log-Logistic 1
distribution has mode: mode of Log-Logistic 1
distribution has variance: variance of Log-Logistic 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: scale of Log-Logistic-1
distribution has parameter: shape of Log-Logistic-1

variance of Gamma 1
type: mathematical object
has name: variance of Gamma 1
has Latex expression: k \theta^2

variance of Standard Two-Sided Power 1
type: mathematical object
has name: variance of Standard Two-Sided Power 1
has Latex expression: \frac{n-2(n-1)m(1-m)}{(n+2)(n+1)^2}

mathematical object
type: Class

distribution has parameter
type: ObjectProperty

PDF of Scaled Inverse Chi-Square
type: probability density function
has name: PDF of Scaled Inverse Chi-Square
has Latex expression: \frac{(\nu/2)^{\nu/2}}{\Gamma(\nu/2)}\sigma^\nu x^{-(\nu/2+1)} \exp\Big(-\frac{1}{2}\nu \sigma^2 \frac{1}{x}\Big)
has R code expression: (nu/2)^(nu/2)/gamma(nu/2)*sigma^nu*x^(-(nu/2+1))*exp(-1/2*nu*sigma^2*1/x)

PDF of Rayleigh 1
type: probability density function
has name: PDF of Rayleigh 1
has Latex expression: \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}
has R code expression: x/sigma^2 * exp(-x^2/(2*sigma^2))

integer
type: Class
subclass of: rational number

relationship between Kumaraswamy 1 and Exponential 1 whereby \\text{If } X \\sim Kumaraswamy11,b \\text{ then } -1-\\logX \\sim Exponential1b
type: Transformation
distribution relationship from: Minimax distribution
distribution relationship to: Exponential 1
distribution relationship definition in Latex: \text{If } X \sim Kumaraswamy1(1,b) \text{ then } -(1-\log(X)) \sim Exponential1(b)
distribution relationship pair expression Latex: Kumaraswamy1(a,b) \rightarrow Exponential1(b)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Kumaraswamy_distribution}

Skew Normal
type: probability distribution
has name: Skew Normal
has code name: SkewNormal1
distribution has PDF: PDF of Skew Normal
distribution has CDF: CDF of Skew Normal
distribution has mean: mean of Skew Normal
distribution has variance: variance of Skew Normal
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: scale of Skew-Normal
distribution has parameter: shape of Skew-Normal
distribution has parameter: location of Skew-Normal

relationship between Log-Normal 3 and Log-Normal 6 whereby m = m, \\sigma_g=\\exp\\sigma
type: Reparameterisation
distribution relationship from: Log-Normal 3
distribution relationship to: lognormal
distribution relationship definition in Latex: m = m, \sigma_g=\exp(\sigma)
distribution relationship pair expression Latex: LogNormal3(m,\sigma) \rightarrow LogNormal6(m,\sigma_g)
distribution relationship reference: ProbOnto spec

CDF of Normal 2
type: cumulative density function
has name: CDF of Normal 2
has Latex expression: \frac12\left[1 + \text{erf}\left( \frac{x-\mu}{\sqrt{v}\sqrt{2}}\right)\right]
has R code expression: 1/2 * (1 + erf((x-mu)/(sqrt(var)*sqrt(2))))

relationship between Beta 1 and Standard Uniform 1 whereby \\alpha=1, \\beta = 1
type: Special case
distribution relationship from: Beta 1
distribution relationship to: Standard Uniform 1
distribution relationship definition in Latex: \alpha=1, \beta = 1
distribution relationship pair expression Latex: Beta1(\alpha,\beta) \rightarrow StandardUniform1
distribution relationship reference: \cite{forbes2011statistical}

inverse scale of Logistic-2
type: mathematical object
has name: inverse scale of Logistic-2
has code name: Logistic2.inverseScale
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau > 0, \tau \in R
has short name: inverse scale
has short code name: inverseScale

probability density function
type: Class
subclass of: mathematical function

CDF of Standard Two-Sided Power 1
type: cumulative density function
has name: CDF of Standard Two-Sided Power 1
has Latex expression: \begin{cases} x^n m^{1-n} &\text{ for } 0 < x < m \\ 1-(1-m) ((1-x)/(1-m))^n &\text{ for } m \leq x < 1 \end{cases}
has R code expression: CDF1 = function(x,m,n) { x^n * m^(1-n) } CDF2 = function(x,m,n) { 1-(1-m)*((1-x)/(1-m))^n }

mean of Beta Negative Binomial1
type: mathematical object
has name: mean of Beta Negative Binomial1
has Latex expression: \begin{cases} \frac{n\beta}{\alpha-1} & \text{if}\ \alpha>1 \\ \infty & \text{otherwise}\ \end{cases}

mode of Wigner Semicircle 1
type: mathematical object
has name: mode of Wigner Semicircle 1
has Latex expression: 0

mean of Standard Two-Sided Power 1
type: mathematical object
has name: mean of Standard Two-Sided Power 1
has Latex expression: \frac{(n-1)m+1}{n+1}

upper bound of Truncated-Normal-1
type: mathematical object
has name: upper bound of Truncated-Normal-1
has code name: TruncatedNormal1.upperBound
has Latex symbol: b
parameter has definition expressed in plain text: b \in R, b > a
has short name: upper bound
has short code name: upperBound

mode of Gamma 1
type: mathematical object
has name: mode of Gamma 1
has Latex expression: (k \,-\, 1)\theta \text{ for } k \;{\geq}\; 1

shape of Lomax-1
type: mathematical object
has name: shape of Lomax-1
has code name: Lomax1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: shape
has short code name: shape

relationship between Log-Normal 1 and Log-Normal 6 whereby m=\\exp\\mu, \\sigma_g=\\exp\\sigma
type: Reparameterisation
distribution relationship from: Log-Normal 1
distribution relationship to: lognormal
distribution relationship definition in Latex: m=\exp(\mu), \sigma_g=\exp(\sigma)
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow LogNormal6(m,\sigma_g)
distribution relationship reference: ProbOnto spec

distribution has median
type: ObjectProperty

event probabilities of Multinomial-1
type: mathematical object
has name: event probabilities of Multinomial-1
has code name: Multinomial1.probabilityOfSuccess
has Latex symbol: p_1, \ldots, p_k
parameter has definition expressed in plain text: p_1, \ldots, p_k, \Sigma p_i = 1
has short name: event probabilities
has short code name: probabilityOfSuccess

CDF of Rayleigh 1
type: cumulative density function
has name: CDF of Rayleigh 1
has Latex expression: 1 - e^{-x^2/(2\sigma^2)}
has R code expression: 1 - exp(-x^2/(2*sigma^2))

Reparameterisation
type: Class
subclass of: relationship between distributions

relationship between Zipf 1 and Uniform Discrete 1 whereby \\alpha = 0, a = 1, b = n
type: Special case
distribution relationship from: Zipf 1
distribution relationship to: Uniform Discrete 1
distribution relationship definition in Latex: \alpha = 0, a = 1, b = n
distribution relationship pair expression Latex: Zipf1(\alpha,n) \rightarrow UniformDiscrete1(a,b)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ZipfDiscreteuniform.pdf}

natural number
type: Class
subclass of: integer

median of Arcsine 2
type: mathematical object
has name: median of Arcsine 2
has Latex expression: (a+b)/2

PDF of Skew Normal
type: probability density function
has name: PDF of Skew Normal
has Latex expression: \frac{1}{\sigma \sqrt{2\pi}} \exp\Big[-\frac{1}{2}\Big(\frac{x-\mu}{\sigma} \Big)^2\Big] \Big[1+\mbox{erf} \Big(\alpha\Big(\frac{x-\mu}{ \sigma\sqrt{2} }\Big)\Big)\Big]
has R code expression: 1/(sigma*sqrt(2*pi)) * exp(-1/2*((x-mu)/sigma)^2) * (1+erf(alpha*((x-mu)/(sigma*sqrt(2)))))

parameterised probability distribution
type: Class
subclass of: probability distribution

relationship between NoncentralT and Student's t-distribution 1 whereby \\delta=0
type: Special case
distribution relationship from: NoncentralT
distribution relationship to: t-distribution
distribution relationship definition in Latex: \delta=0
distribution relationship pair expression Latex: NoncentralT1(\delta,n) \rightarrow StudentT1(n)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NoncentraltT.pdf}

PDF of Normal 2
type: probability density function
has name: PDF of Normal 2
has Latex expression: \frac{1}{\sqrt{v} \sqrt{2 \pi}}e^{-\frac{(x-\mu)^2}{2v}}
has R code expression: 1/(sqrt(var)*sqrt(2*pi))*exp(-(x-mu)^2/(2*var))

PDF of Standard Two-Sided Power 1
type: probability density function
has name: PDF of Standard Two-Sided Power 1
has Latex expression: \begin{cases} n \left(\frac{x}{m}\right)^{n-1} &\text{ for } 0 < x < m \\ n \left(\frac{1-x}{1-m}\right)^{n-1} &\text{ for } m \leq x < 1 \end{cases}
has R code expression: PDF1 = function(x,m,n) { n*(x/m)^(n-1) } PDF2 = function(x,m,n) { n*((1-x)/(1-m))^(n-1) }

PDF of Johnson SU 1
type: probability density function
has name: PDF of Johnson SU 1
has Latex expression: \frac{e^{-\frac12 \left( \gamma + \delta \,\text{arcsinh}\left[\frac{x-\mu}{\sigma}\right]\right)^2} \delta}{\sqrt{2\pi} \sqrt{(x-\mu)^2 + \sigma^2}}
has R code expression: exp(-1/2 * (gamma+delta *asinh((x-mu)/sigma))^2)* delta / (sqrt(2*pi)*sqrt((x-mu)^2 + sigma^2))

shape parameter of Beta-Negative-Binomial1
type: mathematical object
has name: shape parameter of Beta-Negative-Binomial1
has code name: BetaNegativeBinomial1.shape1
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: shape parameter
has short code name: shape1

relationship between Log-Normal 2 and Log-Normal 1 whereby \\mu = \\mu, \\sigma = \\sqrt{v}
type: Reparameterisation
distribution relationship from: Log-Normal 2
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \mu = \mu, \sigma = \sqrt{v}
distribution relationship pair expression Latex: LogNormal2(\mu,v) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

lower bound of Truncated-Normal-1
type: mathematical object
has name: lower bound of Truncated-Normal-1
has code name: TruncatedNormal1.lowerBound
has Latex symbol: a
parameter has definition expressed in plain text: a \in R
has short name: lower bound
has short code name: lowerBound

Epanechnikov 1
type: probability distribution
has name: Epanechnikov 1
has code name: Epanechnikov1
distribution has PDF: PDF of Epanechnikov 1
distribution has CDF: CDF of Epanechnikov 1
distribution has mean: mean of Epanechnikov 1
distribution has variance: variance of Epanechnikov 1
has variate symbol in Latex: x
has Latex expression of support: x \in [-1,1]

relationship between Log-Normal 4 and Log-Normal 1 whereby \\mu = \\logm, \\sigma = \\sqrt{\\logcv^2+1}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \mu = \log(m), \sigma = \sqrt{\log(cv^2+1)}
distribution relationship pair expression Latex: LogNormal4(m,cv) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

Triangular 1
type: probability distribution
has name: Triangular 1
has code name: Triangular1
distribution has PDF: PDF of Triangular 1
distribution has CDF: CDF of Triangular 1
distribution has mean: mean of Triangular 1
distribution has mode: mode of Triangular 1
distribution has variance: variance of Triangular 1
has variate symbol in Latex: x
has Latex expression of support: a \leq x \leq b
distribution has parameter: lower limit of Triangular-1
distribution has parameter: upper limit of Triangular-1
distribution has parameter: shape mode of Triangular-1

t-distribution
type: probability distribution
label: Student's t-distribution 1
has name: Student's t-distribution 1
has code name: StudentT1
distribution has PDF: PDF of Student's t-distribution 1
distribution has CDF: CDF of Student's t-distribution 1
distribution has mean: mean of Student's t-distribution 1
distribution has median: median of Student's t-distribution 1
distribution has mode: mode of Student's t-distribution 1
distribution has variance: variance of Student's t-distribution 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: degrees of freedom of Student-s-t-distribution-1

distribution has mean
type: ObjectProperty

mean of Rayleigh 1
type: mathematical object
has name: mean of Rayleigh 1
has Latex expression: \sigma \sqrt{\frac{\pi}{2}}

median of Gamma 1
type: mathematical object
has name: median of Gamma 1
has Latex expression: \text{No simple closed form}

CDF of Log-Logistic 1
type: cumulative density function
has name: CDF of Log-Logistic 1
has Latex expression: \frac{1}{1+(x/\alpha)^{-\beta}}
has R code expression: 1 / (1+(x/alpha)^(-beta))

Special case
type: Class
subclass of: relationship between distributions

parametric probability distribution
type: Class
subclass of: probability distribution

parameter has definition expressed in plain text
type: DatatypeProperty

PDF of Weibull 1
type: probability density function
has name: PDF of Weibull 1
has R code expression: k/lambda * (x/lambda)^(k-1) * exp(-(x/lambda)^k)

mode of Laplace 1
type: mathematical object
has name: mode of Laplace 1
has Latex expression: \mu

mean of Wigner Semicircle 1
type: mathematical object
has name: mean of Wigner Semicircle 1
has Latex expression: 0

Minimax distribution
type: probability distribution
label: Kumaraswamy 1
has name: Kumaraswamy 1
has code name: Kumaraswamy1
distribution has PDF: PDF of Kumaraswamy 1
distribution has CDF: CDF of Kumaraswamy 1
distribution has SF: SF of Kumaraswamy 1
distribution has HF: HF of Kumaraswamy 1
distribution has mean: mean of Kumaraswamy 1
distribution has median: median of Kumaraswamy 1
distribution has mode: mode of Kumaraswamy 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,1]
distribution has parameter: shape parameter of Kumaraswamy-1

PDF of Log-Normal 6
type: probability density function
has name: PDF of Log-Normal 6
has Latex expression: \frac{1}{x\log(\sigma_g)\sqrt{2 \pi}} \exp\Big[ \frac{-[\log(x/m)]^2}{2 \log^2(\sigma_g)}\Big]
has R code expression: 1/(x*log(sigma_g)*sqrt(2*pi))*exp(-(log(x/m))^2/(2*log(sigma_g)^2))

mode of Student's t-distribution 1
type: mathematical object
has name: mode of Student's t-distribution 1
has Latex expression: 0

relationship between Power-Normal 1 and Log-Normal 1 whereby PowerNormal1 \\text{ distribution converges to LogNormal1 one when } \\lambda \\rightarrow 0
type: Special case
distribution relationship from: Power-Normal 1
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: PowerNormal1 \text{ distribution converges to LogNormal1 one when } \lambda \rightarrow 0
distribution relationship pair expression Latex: PowerNormal1(\lambda,\mu,\sigma) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: \cite{LavielleBook:2014}

variance of Negative Binomial 1
type: mathematical object
has name: variance of Negative Binomial 1
has Latex expression: \frac{r(1-p)}{p^2}

mean of logx of Log-Normal-5
type: mathematical object
has name: mean of log(x) of Log-Normal-5
has code name: LogNormal5.meanLog
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean of log(x)
has short code name: meanLog

shape of Log-Normal-3
type: mathematical object
has name: shape of Log-Normal-3
has code name: LogNormal3.stdevLog
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: shape
has short code name: stdevLog

variance of Arcsine 1
type: mathematical object
has name: variance of Arcsine 1
has Latex expression: \frac{1}{8}

PDF of Uniform 1
type: probability density function
has name: PDF of Uniform 1
has Latex expression: \begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases}
has R code expression: 1/(b-a)

mean of Gamma 1
type: mathematical object
has name: mean of Gamma 1
has Latex expression: k \theta

non numerical set
type: Class

mode of Scaled Inverse Chi-Square
type: mathematical object
has name: mode of Scaled Inverse Chi-Square
has Latex expression: \frac{\nu \sigma^2}{\nu + 2}

CDF of Normal 1
type: cumulative density function
has name: CDF of Normal 1
has Latex expression: \frac12\left[1 + \text{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
has R code expression: 1/2 * (1 + erf((x-mu)/(sigma*sqrt(2))))

shape of Log-Logistic-2
type: mathematical object
has name: shape of Log-Logistic-2
has code name: LogLogistic2.shape
has Latex symbol: \kappa
parameter has definition expressed in plain text: \kappa > 0
has short name: shape
has short code name: shape

median of Rayleigh 1
type: mathematical object
has name: median of Rayleigh 1
has Latex expression: \sigma \sqrt{\log(4)}

variance of Standard Cauchy 1
type: mathematical object
has name: variance of Standard Cauchy 1
has Latex expression: undefined

PMF of Weibull Discrete 1
type: probability mass function
has name: PMF of Weibull Discrete 1
has Latex expression: (1-p)^{x^\beta} - (1-p)^{(x+1)^\beta}
has R code expression: (1-p)^(x^beta) - (1-p)^((x+1)^beta)

relationship between Binomial 1 and Binomial 2 whereby \\alpha = \\logp/1-p
type: Reparameterisation
distribution relationship from: Binomial 1
distribution relationship to: Binomial distribution with logit parameterisation
distribution relationship definition in Latex: \alpha = \log(p/(1-p))
distribution relationship pair expression Latex: Binomial1(n,p) \rightarrow Binomial2(n,\alpha)
distribution relationship reference: ProbOnto spec

median of Laplace 1
type: mathematical object
has name: median of Laplace 1
has Latex expression: \mu

Weibull 1
type: probability distribution
has name: Weibull 1
has code name: Weibull1
distribution has PDF: PDF of Weibull 1
distribution has CDF: CDF of Weibull 1
distribution has SF: SF of Weibull 1
distribution has HF: HF of Weibull 1
distribution has mean: mean of Weibull 1
distribution has median: median of Weibull 1
distribution has mode: mode of Weibull 1
distribution has variance: variance of Weibull 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: shape of Weibull-1
distribution has parameter: scale of Weibull-1

relationship between Gamma 1 and Erlang 1 whereby k \\in N, k=c, \\theta=b
type: Special case
distribution relationship from: Gamma 1
distribution relationship to: Erlang 1
distribution relationship definition in Latex: k \in N, k=c, \theta=b
distribution relationship pair expression Latex: Gamma1(k,\theta) \rightarrow Erlang1(b,c)
distribution relationship reference: \cite{forbes2011statistical} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaErlang.pdf}

parameter has definition
type: DatatypeProperty

probability distribution
type: Class
subclass of: mathematical object

mode of Arcsine 1
type: mathematical object
has name: mode of Arcsine 1
has Latex expression: x \in \{0,1\}

variance of Student's t-distribution 1
type: mathematical object
has name: variance of Student's t-distribution 1
has Latex expression: \begin{cases} \frac{\nu}{\nu - 2} & \text{for }\nu > 2 \\ \infty & \text{for } 1< \nu \leq 2 \\ undefined & \text{else} \end{cases}

PDF of Wiener Diffusion Model 1
type: probability density function
has name: PDF of Wiener Diffusion Model 1
has Latex expression: \frac{\alpha}{(x-\tau)^{3/2}}\exp\Big(-\delta\alpha\beta - \frac{\delta^2(x-\tau)}{2}\Big) \sum^\infty_{k=-\infty}(2k+\beta)\phi\Big(\alpha\frac{2k + \beta}{\sqrt{x-\tau}}\Big)

median of Wigner Semicircle 1
type: mathematical object
has name: median of Wigner Semicircle 1
has Latex expression: 0

lognormal
type: probability distribution
label: Galton
label: Log-Normal 6
has name: Log-Normal 6
has code name: LogNormal6
distribution has PDF: PDF of Log-Normal 6
distribution has CDF: CDF of Log-Normal 6
distribution has mean: mean of Log-Normal 6
distribution has median: median of Log-Normal 6
distribution has mode: mode of Log-Normal 6
distribution has variance: variance of Log-Normal 6
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Log-Normal-6
distribution has parameter: median / geometric mean of Log-Normal-6

mode of Negative Binomial 1
type: mathematical object
has name: mode of Negative Binomial 1
has Latex expression: \lfloor \frac{(1-p)(r-1)}{p} \rfloor

relationship between Muth 1 and Exponential 2 whereby \\text{As } \\kappa \\rightarrow 0 \\text{ for a Muth1}\\kappa\\text{ , the limiting distribution is Exponential2 with mean 1.}
type: Limiting
distribution relationship from: Muth 1
distribution relationship to: Negative exponential
distribution relationship definition in Latex: \text{As } \kappa \rightarrow 0 \text{ for a Muth1(}\kappa\text{) , the limiting distribution is Exponential2 with mean 1.}
distribution relationship pair expression Latex: Muth1(\kappa) \rightarrow Exponential2(\beta)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/MuthExponential.pdf}

precision of Log-Normal-5
type: mathematical object
has name: precision of Log-Normal-5
has code name: LogNormal5.precision
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau > 0
has short name: precision
has short code name: precision

Rician
type: probability distribution
label: Rice 1
has name: Rice 1
has code name: Rice1
distribution has PDF: PDF of Rice 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: noncentrality parameter of Rice-1
distribution has parameter: scale parameter of Rice-1

Uniform 1
type: probability distribution
label: Rectangular Continuous
has name: Uniform 1
has code name: Uniform1
distribution has PDF: PDF of Uniform 1
distribution has CDF: CDF of Uniform 1
distribution has mean: mean of Uniform 1
distribution has median: median of Uniform 1
distribution has mode: mode of Uniform 1
distribution has variance: variance of Uniform 1
has variate symbol in Latex: x
has Latex expression of support: x \in [a,b]
distribution has parameter: minimum of Uniform-1
distribution has parameter: maximum of Uniform-1

variance of Scaled Inverse Chi-Square
type: mathematical object
has name: variance of Scaled Inverse Chi-Square
has Latex expression: \frac{2\nu^2 \sigma^4}{(\nu - 2)^2(\nu-4)} \text{ for } \nu>4

CDF of Gamma 1
type: cumulative density function
has name: CDF of Gamma 1
has Latex expression: \frac{1}{\Gamma(k)} \gamma\left(k,\, \frac{x}{\theta}\right)
has R code expression: 1/gamma(k) * Igamma(k,x/theta)

finite set
type: Class
subclass of: mathematical set

CDF of Weibull Discrete 1
type: cumulative density function
has name: CDF of Weibull Discrete 1
has Latex expression: 1-(1-p)^{(x+1)^\beta}
has R code expression: 1-(1-p)^((x+1)^beta)

Wiener Diffusion Model 1
type: probability distribution
has name: Wiener Diffusion Model 1
has code name: WienerDiffusionModel1
distribution has PDF: PDF of Wiener Diffusion Model 1
has variate symbol in Latex: x
distribution has parameter: boundary separation of Wiener-Diffusion-Model-1
distribution has parameter: drift rate of Wiener-Diffusion-Model-1
distribution has parameter: initial bias of Wiener-Diffusion-Model-1
distribution has parameter: nondecision time of Wiener-Diffusion-Model-1

mode of Rayleigh 1
type: mathematical object
has name: mode of Rayleigh 1
has Latex expression: \sigma

PDF of Normal 1
type: probability density function
has name: PDF of Normal 1
has Latex expression: \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}
has R code expression: 1/(sigma*sqrt(2*pi))*exp(-(x-mu)^2/(2*sigma^2))

mode of Standard Cauchy 1
type: mathematical object
has name: mode of Standard Cauchy 1
has Latex expression: 0

scale of Log-Logistic-2
type: mathematical object
has name: scale of Log-Logistic-2
has code name: LogLogistic2.scale
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0
has short name: scale
has short code name: scale

CDF of Uniform Discrete 1
type: cumulative density function
has name: CDF of Uniform Discrete 1
has Latex expression: (k-a+1)/(b-a+1)
has R code expression: (k-a+1)/(b-a+1)

PDF of Wigner Semicircle 1
type: probability density function
has name: PDF of Wigner Semicircle 1
has Latex expression: \frac2{\pi R^2} \sqrt{R^2-x^2}
has R code expression: 2/(pi * R^2) * sqrt(R^2-x^2)

variance of Log-Logistic 2
type: mathematical object
has name: variance of Log-Logistic 2
has Latex expression: \frac{\pi \big( 2\kappa [1 - \cos(\frac{\pi}{\kappa})^2] + \pi \sin\big(\frac{\pi(\kappa+2)}{\kappa}\big)\big)}{\sin\big(\frac{\pi(\kappa+2)}{\kappa}\big)\big(\cos^2(\frac{\pi}{\kappa}) -1 \big) (\lambda \kappa)^2}

PDF of Rice 1
type: probability density function
has name: PDF of Rice 1
has Latex expression: \frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)
has R code expression: x/sigma^2*exp(-(x^2+nu^2)/2/sigma^2) * besselI(x*nu/sigma^2, 0, expon.scaled = FALSE)

location of Laplace-1
type: mathematical object
has name: location of Laplace-1
has code name: Laplace1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

mode of Log-Normal 5
type: mathematical object
has name: mode of Log-Normal 5
has Latex expression: e^{\mu - \frac{1}{\tau}}

relationship between Logistic 2 and Logistic 1 whereby s=1/\\tau
type: Reparameterisation
distribution relationship from: Logistic 2
distribution relationship to: Logistic 1
distribution relationship definition in Latex: s=1/\tau
distribution relationship pair expression Latex: Logistic2(\mu,\tau) \rightarrow Logistic1(\mu,s)
distribution relationship reference: ProbOnto spec

PDF of Arcsine 2
type: probability density function
has name: PDF of Arcsine 2
has Latex expression: \frac{1}{\pi\sqrt{(x-a)(b-x)}}
has R code expression: 1/pi/sqrt((x-a)*(b-x))

relationship between Power 1 and Standard Power 1 whereby \\text{The StandardPower1}\\beta \\text{ distribution is a special case of the Power1}\\alpha,\\beta \\text{ distribution when } \\alpha = 1.
type: Special case
distribution relationship from: Power 1
distribution relationship to: Standard Power 1
distribution relationship definition in Latex: \text{The StandardPower1}(\beta) \text{ distribution is a special case of the Power1}(\alpha,\beta) \text{ distribution when } \alpha = 1.
distribution relationship pair expression Latex: Power1(\alpha,\beta) \rightarrow StandardPower1(\beta)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PowerStandardpower.pdf}

relationship between Negative Binomial 4 and Negative Binomial 5 whereby \\alpha = r, \\beta = 1-p / p
type: Reparameterisation
distribution relationship from: Negative Binomial 4
distribution relationship to: Negative Binomial 5
distribution relationship definition in Latex: \alpha = r, \beta = (1-p) / p
distribution relationship pair expression Latex: NegativeBinomial4(r,p) \rightarrow NegativeBinomial5(\alpha, \beta)
distribution relationship reference: ProbOnto spec

location of minimum of Frechet-2
type: mathematical object
has name: location of minimum of Frechet-2
has code name: Frechet2.locationOfMinimum
has Latex symbol: m
parameter has definition expressed in plain text: m \in (-\infty, \infty)
has short name: location of minimum
has short code name: locationOfMinimum

PROB k0001431
type: Transformation
label: relationship between Normal 1 and Folded Normal 1 whereby \\text{Given a normally distributed random variable X with mean } \\mu \\text{ and variance } \\sigma^2, \\text{ the random}\\\\ \\text{variable } Y = |X| \\text{ has a folded normal distribution}
distribution relationship from: Normal
distribution relationship to: Folded Normal 1
distribution relationship definition in Latex: \text{Given a normally distributed random variable X with mean } \mu \text{ and variance } \sigma^2, \text{ the random}\\ \text{variable } Y = |X| \text{ has a folded normal distribution}
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow FoldedNormal1(\mu,\sigma)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Folded_normal_distribution}

CDF of Scaled Inverse Chi-Square
type: cumulative density function
has name: CDF of Scaled Inverse Chi-Square
has Latex expression: \Gamma\Big(\frac{\nu}{2},\frac{\sigma^2\nu}{2x}\Big)\Big/\Gamma\Big(\frac{\nu}{2}\Big)
has R code expression: Igamma(nu/2,sigma^2*nu/2/x, lower=F) / gamma(nu/2) \text{ with } Igamma(a, x, lower=F)

relationship between Normal 1 and Standard Normal 1 whereby \\mu = 0, \\sigma = 1
type: Special case
distribution relationship from: Normal
distribution relationship to: Standard Normal 1
distribution relationship definition in Latex: \mu = 0, \sigma = 1
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow StandardNormal1(0,1)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalStandardnormalT.pdf}

mathematical set
type: Class
subclass of: mathematical object

PDF of Gamma 1
type: probability density function
has name: PDF of Gamma 1
has Latex expression: \frac{1}{\Gamma(k) \theta^k} x^{k \,-\, 1} e^{-\frac{x}{\theta}}
has R code expression: 1 / (gamma(k) * theta^k) * x^(k-1) * exp(-x/theta)

vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-2
type: mathematical object
has name: vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-2
has code name: MultivariateGaussianProcess2.inverseScale
has Latex symbol: w
parameter has definition expressed in plain text: w \in R^K, K \in N
has short name: vector of positive inverse scales
has short code name: inverseScale

distribution has nth parameter
type: ObjectProperty

SF of Geometric 1
type: survival function
has name: SF of Geometric 1
has Latex expression: (1 - p)^k
has R code expression: (1 - p)^k

median of Normal 1
type: mathematical object
has name: median of Normal 1
has Latex expression: \mu

Wigner Semicircle 1
type: probability distribution
has name: Wigner Semicircle 1
has code name: WignerSemicircle1
distribution has PDF: PDF of Wigner Semicircle 1
distribution has CDF: CDF of Wigner Semicircle 1
distribution has mean: mean of Wigner Semicircle 1
distribution has median: median of Wigner Semicircle 1
distribution has mode: mode of Wigner Semicircle 1
distribution has variance: variance of Wigner Semicircle 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-R,R)
distribution has parameter: radius of Wigner-Semicircle-1

median of Standard Cauchy 1
type: mathematical object
has name: median of Standard Cauchy 1
has Latex expression: 0

mean of Student's t-distribution 1
type: mathematical object
has name: mean of Student's t-distribution 1
has Latex expression: \begin{cases} 0 & \text{for }\nu > 1 \\ undefined & \text{else} \end{cases}

variance of Rayleigh 1
type: mathematical object
has name: variance of Rayleigh 1
has Latex expression: \frac{4 - \pi}{2} \sigma^2

median of Log-Logistic 2
type: mathematical object
has name: median of Log-Logistic 2
has Latex expression: 1/\lambda

relationship between Negative Binomial 3 and Negative Binomial 6 whereby \\eta = \\log\\mu
type: Reparameterisation
distribution relationship from: Negative Binomial 3
distribution relationship to: Negative Binomial 6
distribution relationship definition in Latex: \eta = \log(\mu)
distribution relationship pair expression Latex: NegativeBinomial3(\mu,\phi) \rightarrow NegativeBinomial6(\eta,\phi)
distribution relationship reference: \cite{stan-manual:2015b}

boundary separation of Wiener-Diffusion-Model-1
type: mathematical object
has name: boundary separation of Wiener-Diffusion-Model-1
has code name: WienerDiffusionModel1.boundSeparation
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R^+
has short name: boundary separation
has short code name: boundSeparation

CDF of Wigner Semicircle 1
type: cumulative density function
has name: CDF of Wigner Semicircle 1
has Latex expression: \frac12 \frac{x\sqrt{R^2-x^2}}{\pi R^2} + \frac{\arcsin(\frac{x}{R})}{\pi}
has R code expression: 1/2 * (x*sqrt(R^2-x^2))/(pi * R^2) + asin(x/R)/pi

median of Student's t-distribution 1
type: mathematical object
has name: median of Student's t-distribution 1
has Latex expression: 0

variance of Laplace 1
type: mathematical object
has name: variance of Laplace 1
has Latex expression: 2 b^2

relationship between Uniform 1 and Standard Uniform 1 whereby a = 0, b = 1
type: Special case
distribution relationship from: Uniform 1
distribution relationship to: Standard Uniform 1
distribution relationship definition in Latex: a = 0, b = 1
distribution relationship pair expression Latex: Uniform1(a,b) \rightarrow StandardUniform1(0,1)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/UniformStandarduniform.pdf}

Arcsine 2
type: probability distribution
has name: Arcsine 2
has code name: Arcsine2
distribution has PDF: PDF of Arcsine 2
distribution has CDF: CDF of Arcsine 2
distribution has mean: mean of Arcsine 2
distribution has median: median of Arcsine 2
distribution has mode: mode of Arcsine 2
distribution has variance: variance of Arcsine 2
has variate symbol in Latex: x
has Latex expression of support: x \in (a,b)
distribution has parameter: upper bound of Arcsine-2
distribution has parameter: lower bound of Arcsine-2

variance of Log-Normal 5
type: mathematical object
has name: variance of Log-Normal 5
has Latex expression: e^{2\mu + \frac{1}{\tau}}[e^{\frac{1}{\tau}}-1]

relationship between Two-Sided Power 1 and Standard Two-Sided Power 1 whereby a=0, b=1
type: Special case
distribution relationship from: Two-Sided Power 1
distribution relationship to: Standard Two-Sided Power 1
distribution relationship definition in Latex: a=0, b=1
distribution relationship pair expression Latex: TwoSidedPower1(a,b,m,n) \rightarrow StandardTwoSidedPower1(m,n)
distribution relationship reference: \cite{van2002standard}

relationship between Standard Normal 1 and Normal 1 whereby X \\sim StandardNormal1 \\text{ and } Y = \\mu + \\sigma X \\Rightarrow Y \\sim Normal1
type: Transformation
distribution relationship from: Standard Normal 1
distribution relationship to: Normal
distribution relationship definition in Latex: X \sim StandardNormal1 \text{ and } Y = \mu + \sigma X \Rightarrow Y \sim Normal1
distribution relationship pair expression Latex: StandardNormal1(0,1) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardnormalNormal.pdf}

scale of Frechet-2
type: mathematical object
has name: scale of Frechet-2
has code name: Frechet2.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma \in (0, \infty)
has short name: scale
has short code name: scale

relationship between Standard Power 1 and Standard Uniform 1 whereby \\text{StandardUniform1 distribution is a special case of the StandardPower1}\\beta \\text{ distribution when } \\beta=1
type: Special case
distribution relationship from: Standard Power 1
distribution relationship to: Standard Uniform 1
distribution relationship definition in Latex: \text{StandardUniform1 distribution is a special case of the StandardPower1}(\beta) \text{ distribution when } \beta=1
distribution relationship pair expression Latex: StandardPower1(\beta) \rightarrow StandardUniform1
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardpowerStandarduniform.pdf}

mean of Scaled Inverse Chi-Square
type: mathematical object
has name: mean of Scaled Inverse Chi-Square
has Latex expression: \frac{\nu \sigma^2}{\nu - 2} \text{ for } \nu>2

Gamma 1
type: probability distribution
has name: Gamma 1
has code name: Gamma1
distribution has PDF: PDF of Gamma 1
distribution has CDF: CDF of Gamma 1
distribution has SF: SF of Gamma 1
distribution has mean: mean of Gamma 1
distribution has median: median of Gamma 1
distribution has mode: mode of Gamma 1
distribution has variance: variance of Gamma 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Gamma-1
distribution has parameter: scale of Gamma-1

relationship between Skew Normal and Chi-squared 1 whereby \\text{If } X \\sim SkewNormal1\\mu,\\sigma,\\alpha \\text{ then } X^2 \\sim ChiSquared11
type: Transformation
distribution relationship from: Skew Normal
distribution relationship to: Chi-square
distribution relationship definition in Latex: \text{If } X \sim SkewNormal1(\mu,\sigma,\alpha) \text{ then } X^2 \sim ChiSquared1(1)
distribution relationship pair expression Latex: SkewNormal1(\mu,\sigma,\alpha) \rightarrow ChiSquared1(k)
distribution relationship reference: \url{http://azzalini.stat.unipd.it/SN/Intro/intro.html}

lower triangular matrix of Multivariate-Gaussian-Process-Distribution-2
type: mathematical object
has name: lower triangular matrix of Multivariate-Gaussian-Process-Distribution-2
has code name: MultivariateGaussianProcess2.choleskyFactor
has Latex symbol: L
parameter has definition expressed in plain text: L \in R^{N \times N}, \text{ lower triangular and such that } LL^T \text{ is positive kernel definite}, \text{with } N \in N
has short name: lower triangular matrix
has short code name: choleskyFactor

numerical set
type: Class

mean of Standard Cauchy 1
type: mathematical object
has name: mean of Standard Cauchy 1
has Latex expression: undefined

mean of Normal 1
type: mathematical object
has name: mean of Normal 1
has Latex expression: \mu

has short code name
type: DatatypeProperty

Weibull Discrete 1
type: probability distribution
has name: Weibull Discrete 1
has code name: WeibullDiscrete1
distribution has PMF: PMF of Weibull Discrete 1
distribution has CDF: CDF of Weibull Discrete 1
distribution has SF: SF of Weibull Discrete 1
distribution has HF: HF of Weibull Discrete 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,\dots\}
distribution has parameter: shape parameter of Weibull-Discrete-1

distribution has relation of type to
type: ObjectProperty

scale of Rayleigh-1
type: mathematical object
has name: scale of Rayleigh-1
has code name: Rayleigh1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: scale
has short code name: scale

mode of Logit Normal 1
type: mathematical object
has name: mode of Logit Normal 1
has Latex expression: \text{ no analytical solution }

CDF of Gumbel 1
type: cumulative density function
has name: CDF of Gumbel 1
has Latex expression: e^{-e^{-(x-\mu)/\beta}}
has R code expression: exp(-exp(-(x-mu)/beta))

relationship between Chi-squared 1 and Exponential 1 whereby k=2 \\text{ and } \\lambda=1/2
type: Special case
distribution relationship from: Chi-square
distribution relationship to: Exponential 1
distribution relationship definition in Latex: k=2 \text{ and } \lambda=1/2
distribution relationship pair expression Latex: ChiSquared1(k) \rightarrow Exponential1(\lambda)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ChisquareExponential.pdf}

mean of Weibull 1
type: mathematical object
has name: mean of Weibull 1
has Latex expression: \lambda \, \Gamma(1+1/k)

SF of Standard Cauchy 1
type: survival function
has name: SF of Standard Cauchy 1
has Latex expression: \frac{\pi - 2\arctan(x)}{2\pi}
has R code expression: (pi - 2*atan(x))/2/pi

relationship between Log-Normal 1 and Log-Normal 4 whereby m = \\exp\\mu, cv = \\sqrt{\\exp\\sigma^2 - 1}
type: Reparameterisation
distribution relationship from: Log-Normal 1
distribution relationship to: lognormal
distribution relationship definition in Latex: m = \exp(\mu), cv = \sqrt{\exp(\sigma^2) - 1}
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow LogNormal4(m,cv)
distribution relationship reference: ProbOnto spec

population size of Hypergeometric-1
type: mathematical object
has name: population size of Hypergeometric-1
has code name: Hypergeometric1.populationSize
has Latex symbol: N
parameter has definition expressed in plain text: N \in \left\{0,1,2,\dots\right\}
has short name: population size
has short code name: populationSize

CDF of Truncated Normal 1
type: cumulative density function
has name: CDF of Truncated Normal 1
has Latex expression: \frac{\Phi(\frac{x-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}
has R code expression: ( Phi((x-mu)/sigma)-Phi((a-mu)/sigma) ) / ( Phi((b-mu)/sigma)-Phi((a-mu)/sigma) ) Phi = function(x) { 1/2 * (1 + erf(x/(sqrt(2)))) } erf = function(x) { 2 * pnorm(x * sqrt(2)) - 1 }

number of trials of Beta-binomial-1
type: mathematical object
has name: number of trials of Beta-binomial-1
has code name: BetaBinomial1.numberOfTrials
has Latex symbol: n
parameter has definition expressed in plain text: n \in N, n > 0
has short name: number of trials
has short code name: numberOfTrials

variance of Beta-binomial 1
type: mathematical object
has name: variance of Beta-binomial 1
has Latex expression: \frac{n\alpha\beta(\alpha+\beta+n)}{(\alpha+\beta)^2(\alpha+\beta+1)}

mode of Multivariate Normal 3
type: mathematical object
has name: mode of Multivariate Normal 3
has Latex expression: \mu

relationship between Negative Binomial 1 and Geometric 1 whereby n=1
type: Special case
distribution relationship from: Negative Binomial 1
distribution relationship to: Geometric 1
distribution relationship definition in Latex: n=1
distribution relationship pair expression Latex: NegativeBinomial1(r,p) \rightarrow Geometric1(p)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalGeometric.pdf}

PMF of Negative Binomial 1
type: probability mass function
has name: PMF of Negative Binomial 1
has Latex expression: \binom {k+r-1}k p^r (1-p)^k
has R code expression: choose(k+r-1,k) * p^r * (1-p)^k

scale of Student-s-t-distribution-2
type: mathematical object
has name: scale of Student-s-t-distribution-2
has code name: StudentT2.scale
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau > 0
has short name: scale
has short code name: scale

mode of Uniform 1
type: mathematical object
has name: mode of Uniform 1
has Latex expression: \text{any value in }[a,b]

NA of Makeham-1
type: mathematical object
has name: NA of Makeham-1
has code name: Makeham1.gamma
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma > 0
has short code name: gamma

relationship between Log-Normal 1 and Log-Normal 2 whereby \\mu = \\mu, v = \\sigma^2
type: Reparameterisation
distribution relationship from: Log-Normal 1
distribution relationship to: Log-Normal 2
distribution relationship definition in Latex: \mu = \mu, v = \sigma^2
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow LogNormal2(\mu,v)
distribution relationship reference: ProbOnto spec

variance of Multivariate Normal 2
type: mathematical object
has name: variance of Multivariate Normal 2
has Latex expression: T^{-1}

mode of Log-Normal 6
type: mathematical object
has name: mode of Log-Normal 6
has Latex expression: m / e^{\log^2(\sigma_g)}

SF of Kumaraswamy 1
type: survival function
has name: SF of Kumaraswamy 1
has Latex expression: (1-x^a)^b
has R code expression: -(1-x^a)^b

Noncentral chi-squared 1
type: probability distribution
has name: Noncentral chi-squared 1
has code name: NoncentralChiSquared1
distribution has PDF: PDF of Noncentral chi-squared 1
distribution has CDF: CDF of Noncentral chi-squared 1
distribution has mean: mean of Noncentral chi-squared 1
distribution has variance: variance of Noncentral chi-squared 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: noncentrality parameter of Noncentral-chi-squared-1
distribution has parameter: degrees of freedom of Noncentral-chi-squared-1

Uniform Discrete 1
type: probability distribution
label: Rectangular Discrete 1
has name: Uniform Discrete 1
has code name: UniformDiscrete1
distribution has PMF: PMF of Uniform Discrete 1
distribution has CDF: CDF of Uniform Discrete 1
distribution has mean: mean of Uniform Discrete 1
distribution has median: median of Uniform Discrete 1
distribution has variance: variance of Uniform Discrete 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{a,a+1,...,b-1,b\}
distribution has parameter: minimum of Uniform-Discrete-1
distribution has parameter: maximum of Uniform-Discrete-1

initial bias of Wiener-Diffusion-Model-1
type: mathematical object
has name: initial bias of Wiener-Diffusion-Model-1
has code name: WienerDiffusionModel1.initialBias
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta \in [0,1]
has short name: initial bias
has short code name: initialBias

mean of Standard Logistic 1
type: mathematical object
has name: mean of Standard Logistic 1
has Latex expression: 0

CDF of Log-Normal 7
type: cumulative density function
has name: CDF of Log-Normal 7
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log(\frac{\mu_N}{\sqrt{\sigma_N^2/\mu_N^2 + 1}})}{\sqrt{2 \log(1+\sigma_N^2/\mu_N^2)}}\Big]
has R code expression: 1/2 + 1/2 * erf( (log(x)-log(mu_N/sqrt(sigma_N^2/mu_N^2 + 1))) / sqrt(2 * log(1+sigma_N^2/mu_N^2)) )

variance of Logit Normal 1
type: mathematical object
has name: variance of Logit Normal 1
has Latex expression: \text{ no analytical solution }

variance of Hypergeometric 1
type: mathematical object
has name: variance of Hypergeometric 1
has Latex expression: n{K\over N}{(N-K)\over N}{N-n\over N-1}

mean of Gumbel 1
type: mathematical object
has name: mean of Gumbel 1
has Latex expression: \mu + \beta\,\gamma_E; \text{ with is Euler constant } \gamma_E

HF of Standard Cauchy 1
type: hazard function
has name: HF of Standard Cauchy 1
has Latex expression: \frac2{(1+x^2)(\pi - 2\arctan(x))}
has R code expression: 2/(1+x^2)/(pi - 2*atan(x))

SF of Weibull 1
type: survival function
has name: SF of Weibull 1
has Latex expression: \exp(-(x/\lambda)^k)
has R code expression: exp(-(x/lambda)^k)

relationship between Normal 1 and Normal 3 whereby \\mu = \\mu, \\tau = 1 / \\sigma^2
type: Reparameterisation
distribution relationship from: Normal
distribution relationship to: Gaussian
distribution relationship definition in Latex: \mu = \mu, \tau = 1 / \sigma^2
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow Normal3(\mu,\tau)
distribution relationship reference: ProbOnto spec

PDF of Truncated Normal 1
type: probability density function
has name: PDF of Truncated Normal 1
has Latex expression: \frac{\frac{1}{\sigma} \phi(\frac{x-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}
has R code expression: ( 1/sigma * phi((x-mu)/sigma) ) / ( Phi((b-mu)/sigma)-Phi((a-mu)/sigma) ) phi = function(x) { 1/(sqrt(2*pi))*exp(-x^2/2) } Phi = function(x) { 1/2 * (1 + erf(x/(sqrt(2)))) } erf = function(x) { 2 * pnorm(x * sqrt(2)) - 1 }

relationship between Gamma 1 and Gamma 2 whereby r=k, \\mu = 1/ \\theta
type: Reparameterisation
distribution relationship from: Gamma 1
distribution relationship to: Gamma 2
distribution relationship definition in Latex: r=k, \mu = 1/ \theta
distribution relationship pair expression Latex: Gamma1(k,\theta) \rightarrow Gamma2(r,\mu)
distribution relationship reference: ProbOnto spec

mean of Beta-binomial 1
type: mathematical object
has name: mean of Beta-binomial 1
has Latex expression: \frac{n\alpha}{\alpha+\beta}

CDF of Standard Logistic 1
type: cumulative density function
has name: CDF of Standard Logistic 1
has Latex expression: \frac{e^{x}}{1+e^{x}}
has R code expression: exp(x)/(1+exp(x))

median of Uniform 1
type: mathematical object
has name: median of Uniform 1
has Latex expression: \tfrac{1}{2}(a+b)

Negative Binomial 1
type: probability distribution
has name: Negative Binomial 1
has code name: NegativeBinomial1
distribution has PMF: PMF of Negative Binomial 1
distribution has CDF: CDF of Negative Binomial 1
distribution has mean: mean of Negative Binomial 1
distribution has mode: mode of Negative Binomial 1
distribution has variance: variance of Negative Binomial 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\} \text{ -- number of failures}
distribution has parameter: number of successes of Negative-Binomial-1
distribution has parameter: success probability of Negative-Binomial-1

rate of decay of Conway-Maxwell-Poisson-1
type: mathematical object
has name: rate of decay of Conway-Maxwell-Poisson-1
has code name: ConwayMaxwellPoisson1.rateOfDecay
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu \ge 0
has short name: rate of decay
has short code name: rateOfDecay

degrees of freedom of Student-s-t-distribution-2
type: mathematical object
has name: degrees of freedom of Student-s-t-distribution-2
has code name: StudentT2.degreesOfFreedom
has Latex symbol: k
parameter has definition expressed in plain text: k \ge 2
has short name: degrees of freedom
has short code name: degreesOfFreedom

drift rate of Wiener-Diffusion-Model-1
type: mathematical object
has name: drift rate of Wiener-Diffusion-Model-1
has code name: WienerDiffusionModel1.driftRate
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta \in R
has short name: drift rate
has short code name: driftRate

location of Multivariate-Normal-2
type: mathematical object
has name: location of Multivariate-Normal-2
has code name: MultivariateNormal2.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R^k
has short name: location
has short code name: mean

NA of Makeham-1
type: mathematical object
has name: NA of Makeham-1
has code name: Makeham1.kappa
has Latex symbol: \kappa
parameter has definition expressed in plain text: \kappa > 0
has short code name: kappa

median of Log-Normal 5
type: mathematical object
has name: median of Log-Normal 5
has Latex expression: e^\mu

PDF of Noncentral chi-squared 1
type: probability density function
has name: PDF of Noncentral chi-squared 1
has Latex expression: \sum^\infty_{k=0} \frac{e^{-\delta/2} (\delta/2)^k}{k!} \frac{e^{-x/2} x^{(n+2k)/2-1}}{2^{(n+2k)/2} \Gamma[(n+2k)/2]}
has R code expression: sumFromTo = function(x,delta,n,fromValue,toValue) { PDF=array(0,length(x)); for(j in 1:length(x)) { for(i in fromValue:toValue) { PDF[j] = PDF[j] + exp(-delta/2)*(delta/2)^i/factorial(i) * (exp(-x[j]/2)*x[j]^((n+2*i)/2-1)) / (2^((n+2*i)/2)*gamma((n+2*i)/2)) } } return(PDF) }

HF of Kumaraswamy 1
type: hazard function
has name: HF of Kumaraswamy 1
has Latex expression: \frac{a \,b \,x^{a-1}}{ (1-x^a)}
has R code expression: a*b*x^(a-1) / (1-x^a)

median of Log-Normal 6
type: mathematical object
has name: median of Log-Normal 6
has Latex expression: m

PMF of Uniform Discrete 1
type: probability mass function
has name: PMF of Uniform Discrete 1
has Latex expression: 1/(b-a+1)
has R code expression: 1/(b-a+1)

number of trials of Hypergeometric-1
type: mathematical object
has name: number of trials of Hypergeometric-1
has code name: Hypergeometric1.numberOfTrials
has Latex symbol: n
parameter has definition expressed in plain text: n \in \left\{0,1,2,\dots,N\right\}
has short name: number of trials
has short code name: numberOfTrials

relationship between Zero-Inflated Negative Binomial 1 and Poisson 1 whereby p0=0, \\tau \\rightarrow 0
type: Limiting
distribution relationship from: Zero-Inflated Negative Binomial 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: p0=0, \tau \rightarrow 0
distribution relationship pair expression Latex: ZeroInflatedNegativeBinomial1(\lambda,\tau,p0) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{Troconiz:2009fv}

CDF of Standard Cauchy 1
type: cumulative density function
has name: CDF of Standard Cauchy 1
has Latex expression: \frac{\pi + 2\arctan(x)}{2\pi}
has R code expression: (pi + 2*atan(x))/2/pi

median of Gumbel 1
type: mathematical object
has name: median of Gumbel 1
has Latex expression: \mu - \beta\,\ln(\ln(2))

location of Logit-Normal-1
type: mathematical object
has name: location of Logit-Normal-1
has code name: LogitNormal1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

HF of Weibull Discrete 1
type: hazard function
has name: HF of Weibull Discrete 1
has Latex expression: 1-(1-p)^{(x+1)^\beta-x^\beta}
has R code expression: 1-(1-p)^((x+1)^beta-x^beta)

location of Multivariate-Normal-1
type: mathematical object
has name: location of Multivariate-Normal-1
has code name: MultivariateNormal1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R^k
has short name: location
has short code name: mean

right beta parameter of Beta-binomial-1
type: mathematical object
has name: right beta parameter of Beta-binomial-1
has code name: BetaBinomial1.beta
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: right beta parameter
has short code name: beta

Truncated Normal 1
type: probability distribution
has name: Truncated Normal 1
has code name: TruncatedNormal1
distribution has PDF: PDF of Truncated Normal 1
distribution has CDF: CDF of Truncated Normal 1
distribution has mean: mean of Truncated Normal 1
distribution has variance: variance of Truncated Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in [a,b]
distribution has parameter: upper bound of Truncated-Normal-1
distribution has parameter: lower bound of Truncated-Normal-1
distribution has parameter: standard deviation of Truncated-Normal-1
distribution has parameter: mean of Truncated-Normal-1

relationship between Log-Normal 2 and Log-Normal 3 whereby m = \\exp\\mu, \\sigma = \\sqrt{v}
type: Reparameterisation
distribution relationship from: Log-Normal 2
distribution relationship to: Log-Normal 3
distribution relationship definition in Latex: m = \exp(\mu), \sigma = \sqrt{v}
distribution relationship pair expression Latex: LogNormal2(\mu,v) \rightarrow LogNormal3(m,\sigma)
distribution relationship reference: ProbOnto spec

upper limit of Triangular-1
type: mathematical object
has name: upper limit of Triangular-1
has code name: Triangular1.upperLimit
has Latex symbol: b
parameter has definition expressed in plain text: b \in R, a < b
has short name: upper limit
has short code name: upperLimit

PDF of Log-Normal 7
type: probability density function
has name: PDF of Log-Normal 7
has Latex expression: \frac{1}{x \sqrt{2 \pi \log\Big(1+\sigma_N^2/\mu_N^2\Big)}} \exp\Bigg( \frac{-\Big[ \log(x) - \log\Big(\frac{\mu_N}{\sqrt{1+\sigma_N^2/\mu_N^2}}\Big)\Big]^2}{2\log\Big(1+\sigma_N^2/\mu_N^2\Big)}\Bigg)
has R code expression: 1 / (x * sqrt(2 * pi * log(1+sigma_N^2/mu_N^2 ) )) * exp( -( log(x) - log(mu_N/sqrt(1+sigma_N^2/mu_N^2)) )^2 / (2*log(1+sigma_N^2/mu_N^2)) )

Logistic 2
type: probability distribution
has name: Logistic 2
has code name: Logistic2
distribution has PDF: PDF of Logistic 2
distribution has CDF: CDF of Logistic 2
distribution has mean: mean of Logistic 2
distribution has median: median of Logistic 2
distribution has mode: mode of Logistic 2
distribution has variance: variance of Logistic 2
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: inverse scale of Logistic-2
distribution has parameter: location of Logistic-2

mean of Uniform 1
type: mathematical object
has name: mean of Uniform 1
has Latex expression: \tfrac{1}{2}(a+b)

PDF of Power 1
type: probability density function
has name: PDF of Power 1
has Latex expression: \frac{\beta x^{\beta-1}}{\alpha^\beta}
has R code expression: (beta * x^(beta-1))/alpha^beta

PDF of Standard Logistic 1
type: probability density function
has name: PDF of Standard Logistic 1
has Latex expression: \frac{e^{x}}{\left(1+e^{x}\right)^2}
has R code expression: exp(x) / (1+exp(x))^2

mean of Negative Binomial 1
type: mathematical object
has name: mean of Negative Binomial 1
has Latex expression: \frac{r(1-p)}{p}

mean of Log-Normal 6
type: mathematical object
has name: mean of Log-Normal 6
has Latex expression: m\,e^{\frac{1}{2} \log^2(\sigma_g)}

CDF of Kumaraswamy 1
type: cumulative density function
has name: CDF of Kumaraswamy 1
has Latex expression: 1-(1-x^a)^b
has R code expression: 1-(1-x^a)^b

relationship between Generalized Negative Binomial 1 and Inverse Binomial 1 whereby \\beta=2, \\theta=1-p
type: Reparameterisation
distribution relationship from: Generalized Negative Binomial 1
distribution relationship to: Inverse Binomial 1
distribution relationship definition in Latex: \beta=2, \theta=1-p
distribution relationship pair expression Latex: GeneralizedNegativeBinomial1(\theta,\beta,m) \rightarrow InverseBinomial1(k,p)
distribution relationship reference: \cite{yanagimoto1989inverse}

CDF of Noncentral chi-squared 1
type: cumulative density function
has name: CDF of Noncentral chi-squared 1
has Latex expression: e^{-\delta/2} \sum^\infty_{k=0} \frac{(\delta/2)^k}{k!} Q(x;n+2k)\\ \text{ with } Q(x;p) = \frac{\gamma(p/2,x/2)}{\Gamma(p/2)}
has R code expression: CDF = function(x,delta,n,lowerLimit,upperLimit) { exp(-delta/2) * CDFsum(x,n,delta,0,Ninfty) } CDFsum = function(x,n,delta,fromValue,toValue) { CDF=array(0,length(x)); for(j in 1:length(x)) { for(k in fromValue:toValue) { CDF[j] = CDF[j] + (delta/2)^k/factorial(k) * Igamma((n+2*k)/2,x[j]/2,lower=T)/gamma((n+2*k)/2) } } return(CDF) }

relationship between Log-Normal 2 and Log-Normal 5 whereby \\mu = \\mu, \\tau = 1 / v
type: Reparameterisation
distribution relationship from: Log-Normal 2
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu = \mu, \tau = 1 / v
distribution relationship pair expression Latex: LogNormal2(\mu,v) \rightarrow LogNormal5(\mu,\tau)
distribution relationship reference: ProbOnto spec

mean of Multivariate Normal 2
type: mathematical object
has name: mean of Multivariate Normal 2
has Latex expression: \mu

HF of Weibull 1
type: hazard function
has name: HF of Weibull 1
has Latex expression: \frac{k}\lambda \Big(\frac{x}\lambda\Big)^{k-1}
has R code expression: k/lambda *(x/lambda)^(k-1)

variance of Student's t-distribution 2
type: mathematical object
has name: variance of Student's t-distribution 2
has Latex expression: \frac{1}{\tau} \frac{k}{k-2} \text{ for } k > 2

nondecision time of Wiener-Diffusion-Model-1
type: mathematical object
has name: nondecision time of Wiener-Diffusion-Model-1
has code name: WienerDiffusionModel1.nondecisionTime
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau \in R^+
has short name: nondecision time
has short code name: nondecisionTime

number of successes of Hypergeometric-1
type: mathematical object
has name: number of successes of Hypergeometric-1
has code name: Hypergeometric1.numberOfSuccesses
has Latex symbol: K
parameter has definition expressed in plain text: K \in \left\{0,1,2,\dots,N\right\}
has short name: number of successes
has short code name: numberOfSuccesses

relationship between Log-Logistic 1 and Logistic 1 whereby \\text{If } X \\sim LogLogistic1\\alpha,\\beta \\Rightarrow Y = logX \\sim Logistic1\\mu,s \\text{ with } \\mu=\\log\\alpha, s=1/\\beta
type: Transformation
distribution relationship from: Log-Logistic 1
distribution relationship to: Logistic 1
distribution relationship definition in Latex: \text{If } X \sim LogLogistic1(\alpha,\beta) \Rightarrow Y = log(X) \sim Logistic1(\mu,s) \text{ with } \mu=\log(\alpha), s=1/\beta)
distribution relationship pair expression Latex: LogLogistic1(\alpha,\beta) \rightarrow Logistic1(\mu,s)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Log-logistic_distribution#Related_distributions}

covariance matrix of Multivariate-Normal-1
type: mathematical object
has name: covariance matrix of Multivariate-Normal-1
has code name: MultivariateNormal1.covarianceMatrix
has Latex symbol: \Sigma
parameter has definition expressed in plain text: \Sigma \in R^{k\times k}
has short name: covariance matrix
has short code name: covarianceMatrix

mean of Folded Normal 1
type: mathematical object
has name: mean of Folded Normal 1
has Latex expression: \sigma \sqrt{\tfrac{2}{\pi}} \, e^{(-\mu^2/2\sigma^2)} + \mu \left(1 - 2\,\Phi(\tfrac{-\mu}{\sigma}) \right)

left beta parameter of Beta-binomial-1
type: mathematical object
has name: left beta parameter of Beta-binomial-1
has code name: BetaBinomial1.alpha
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: left beta parameter
has short code name: alpha

vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-1
type: mathematical object
has name: vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-1
has code name: MultivariateGaussianProcess1.inverseScale
has Latex symbol: w
parameter has definition expressed in plain text: w \in R^K, K \in N
has short name: vector of positive inverse scales
has short code name: inverseScale

mode of Gumbel 1
type: mathematical object
has name: mode of Gumbel 1
has Latex expression: \mu

PDF of Standard Cauchy 1
type: probability density function
has name: PDF of Standard Cauchy 1
has Latex expression: \frac1\pi \frac1{1+x^2}
has R code expression: 1/pi/(1+x^2)

lognormal
type: probability distribution
label: Galton
label: Log-Normal 7
has name: Log-Normal 7
has code name: LogNormal7
distribution has PDF: PDF of Log-Normal 7
distribution has CDF: CDF of Log-Normal 7
distribution has mean: mean of Log-Normal 7
distribution has median: median of Log-Normal 7
distribution has mode: mode of Log-Normal 7
distribution has variance: variance of Log-Normal 7
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: standard deviation of Log-Normal-7
distribution has parameter: mean of x of Log-Normal-7

shape mode of Triangular-1
type: mathematical object
has name: shape (mode) of Triangular-1
has code name: Triangular1.shape
has Latex symbol: c
parameter has definition expressed in plain text: c \in R
has short name: shape (mode)
has short code name: shape

Power 1
type: probability distribution
has name: Power 1
has code name: Power1
distribution has PDF: PDF of Power 1
distribution has CDF: CDF of Power 1
distribution has SF: SF of Power 1
distribution has HF: HF of Power 1
distribution has mean: mean of Power 1
distribution has variance: variance of Power 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape parameter of Power-1
distribution has parameter: scale parameter of Power-1

PDF of Logistic 2
type: probability density function
has name: PDF of Logistic 2
has Latex expression: \frac{\tau e^{-\tau(x-\mu)}} {\left(1+e^{-\tau(x-\mu)}\right)^2}
has R code expression: (tau * exp(-tau*(x-mu))) / (1+exp(-tau*(x-mu)))^2

CDF of Uniform 1
type: cumulative density function
has name: CDF of Uniform 1
has Latex expression: \begin{cases} 0 & \text{for } x < a \\ \frac{x-a}{b-a} & \text{for } x \in [a,b) \\ 1 & \text{for } x \ge b \end{cases}
has R code expression: (x-a)/(b-a)

CDF of Negative Binomial 1
type: cumulative density function
has name: CDF of Negative Binomial 1
has Latex expression: 1 - I_{1-p}(k+1,r)
has R code expression: 1 - Rbeta(1-p, k+1, r, lower = T)

Standard Logistic 1
type: probability distribution
has name: Standard Logistic 1
has code name: StandardLogistic1
distribution has PDF: PDF of Standard Logistic 1
distribution has CDF: CDF of Standard Logistic 1
distribution has mean: mean of Standard Logistic 1
distribution has median: median of Standard Logistic 1
distribution has mode: mode of Standard Logistic 1
distribution has variance: variance of Standard Logistic 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)

PDF of Kumaraswamy 1
type: probability density function
has name: PDF of Kumaraswamy 1
has Latex expression: a b x^{a-1} (1-x^a)^{b-1}
has R code expression: a*b*x^(a-1) * (1-x^a)^(b-1)

CDF of Log-Normal 6
type: cumulative density function
has name: CDF of Log-Normal 6
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log m}{\sqrt{2}\log(\sigma_g)}\Big]
has R code expression: 1/2 + 1/2 * erf( (log(x)-log(m)) / (sqrt(2)*log(sigma_g)) )

relationship between Negative Binomial 5 and Negative Binomial 4 whereby r = \\alpha, p = 1 / 1 + \\beta
type: Reparameterisation
distribution relationship from: Negative Binomial 5
distribution relationship to: Negative Binomial 4
distribution relationship definition in Latex: r = \alpha, p = 1 / (1 + \beta)
distribution relationship pair expression Latex: NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial4(r,p)
distribution relationship reference: ProbOnto spec

mean of Student-s-t-distribution-2
type: mathematical object
has name: mean of Student-s-t-distribution-2
has code name: StudentT2.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: mean

CDF of Weibull 1
type: cumulative density function
has name: CDF of Weibull 1
has Latex expression: 1- \exp(-(x/\lambda)^k)
has R code expression: 1- exp(-(x/lambda)^k)

Beta 1
type: probability distribution
has name: Beta 1
has code name: Beta1
distribution has PDF: PDF of Beta 1
distribution has CDF: CDF of Beta 1
distribution has SF: SF of Beta 1
distribution has HF: HF of Beta 1
distribution has mean: mean of Beta 1
distribution has median: median of Beta 1
distribution has mode: mode of Beta 1
distribution has variance: variance of Beta 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,1)
distribution has parameter: shape of Beta-1

mode of Multivariate Normal 2
type: mathematical object
has name: mode of Multivariate Normal 2
has Latex expression: \mu

Breit-Wigner
type: probability distribution
label: Standard Cauchy 1
label: Lorentz
has name: Standard Cauchy 1
has code name: StandardCauchy1
distribution has PDF: PDF of Standard Cauchy 1
distribution has CDF: CDF of Standard Cauchy 1
distribution has SF: SF of Standard Cauchy 1
distribution has HF: HF of Standard Cauchy 1
distribution has mean: mean of Standard Cauchy 1
distribution has median: median of Standard Cauchy 1
distribution has mode: mode of Standard Cauchy 1
distribution has variance: variance of Standard Cauchy 1
has variate symbol in Latex: x
has Latex expression of support: x \in R

PDF of Hyperbolic secant 1
type: probability density function
has name: PDF of Hyperbolic secant 1
has Latex expression: \text{sech}\Big(\pi x\Big)
has R code expression: sech(pi*x)

CDF of Half-normal 1
type: cumulative density function
has name: CDF of Half-normal 1
has Latex expression: \text{erf}(\theta x / \sqrt{\pi})
has R code expression: erf(theta * x / sqrt(pi))

PDF of Logit Normal 1
type: probability density function
has name: PDF of Logit Normal 1
has Latex expression: \frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}}\frac{1}{x (1-x)}
has R code expression: 1/(sigma * sqrt(2*pi)) * exp(-(logit(x) - mu)^2)/(2*sigma^2)) * 1 / (x*(1-x))

variance of Triangular 1
type: mathematical object
has name: variance of Triangular 1
has Latex expression: (a^2 + b^2 + c^2 - ab - ac - bc)/18

PMF of YuleSimon1
type: probability mass function
has name: PMF of YuleSimon1
has Latex expression: \rho B(k, \rho+1)
has R code expression: rho*beta(k,rho+1)

scale of Weibull-1
type: mathematical object
has name: scale of Weibull-1
has code name: Weibull1.scale
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in (0, +\infty)
has short name: scale
has short code name: scale

relationship between Frechet 2 and Weibull 1 whereby X \\sim Frechet2\\alpha,\\sigma,m \\text{ with } m=0 \\rightarrow X^{-1} \\sim Weibull\\lambda=1/\\sigma,k=\\alpha
type: Transformation
distribution relationship from: Inverse Weibull
distribution relationship to: Weibull 1
distribution relationship definition in Latex: X \sim Frechet2(\alpha,\sigma,m) \text{ with } m=0 \rightarrow X^{-1} \sim Weibull(\lambda=1/\sigma,k=\alpha)
distribution relationship pair expression Latex: Frechet2(\alpha,\sigma,m) \rightarrow Weibull1(\lambda,k)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution} \\ \cite{stan-manual:2015b}

mode of Multivariate Normal 1
type: mathematical object
has name: mode of Multivariate Normal 1
has Latex expression: \mu

CDF of Folded Normal 1
type: cumulative density function
has name: CDF of Folded Normal 1
has Latex expression: \frac{1}{2}\left[ \mbox{erf}\left(\frac{x+\mu}{\sigma\sqrt{2}}\right) + \mbox{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
has R code expression: 1/2 * (erf((x-mu)/(sigma*sqrt(2))) + erf((x+mu)/(sigma*sqrt(2))))

has code name
type: DatatypeProperty

standard deviation of Truncated-Normal-1
type: mathematical object
has name: standard deviation of Truncated-Normal-1
has code name: TruncatedNormal1.stdev
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: standard deviation
has short code name: stdev

PDF of Amoroso 1
type: probability density function
has name: PDF of Amoroso 1
has Latex expression: \frac1{\Gamma(\alpha)} \left| \frac{\beta}{\theta}\right|\left(\frac{x-a}{\theta}\right)^{\alpha\beta-1} \exp\left[-\left(\frac{x-a}{\theta}\right)^\beta \right]
has R code expression: 1/gamma(alpha) * abs(beta/theta) * ((x-a)/theta)^(alpha*beta-1) * exp(-((x-a)/theta)^beta)

PDF of Beta 1
type: probability density function
has name: PDF of Beta 1
has Latex expression: \frac{x^{\alpha-1}(1-x)^{\beta-1}} {B(\alpha,\beta)}
has R code expression: (x^(alpha-1)*(1-x)^(beta-1))/beta(alpha,beta)

Johnson SU 1
type: probability distribution
has name: Johnson SU 1
has code name: JohnsonSU1
distribution has PDF: PDF of Johnson SU 1
distribution has CDF: CDF of Johnson SU 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: shape parameter of Johnson-SU-1
distribution has parameter: scale parameter of Johnson-SU-1
distribution has parameter: location parameter of Johnson-SU-1

relationship between Normal 1 and Standard Normal 1 whereby X \\sim Normal1\\mu,\\sigma; Y = X - \\mu / \\sigma; Y \\sim StandardNormal1
type: Transformation
distribution relationship from: Normal
distribution relationship to: Standard Normal 1
distribution relationship definition in Latex: X \sim Normal1(\mu,\sigma); Y = (X - \mu) / \sigma; Y \sim StandardNormal1
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow StandardNormal1(0,1)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalStandardnormalT.pdf}

CDF of Generalized Gamma 2
type: cumulative density function
has name: CDF of Generalized Gamma 2
has Latex expression: \frac{\gamma(c,(\frac{x-a}{b})^k)}{\Gamma(c)}
has R code expression: Igamma(c, ((x-a)/b)^k, lower=T) / gamma(c)

HF of Power 1
type: hazard function
has name: HF of Power 1
has Latex expression: \frac{\beta x^{\beta-1} \alpha^{-\beta}}{\alpha^\beta - x^\beta}
has R code expression: (beta * x^(beta-1) * alpha^(-beta)) / (alpha^beta - x^beta)

PROB k0001412
type: Special case
label: relationship between Amoroso 1 and Frechet 1 whereby \\text{Frechet1}\\alpha,\\sigma \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a=0, \\theta=\\sigma, \\alpha=1, \\beta rightarrow -\\beta
distribution relationship from: Amoroso 1
distribution relationship to: Frechet 1
distribution relationship definition in Latex: \text{Frechet1}(\alpha,\sigma) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a=0, \theta=\sigma, \alpha=1, \beta rightarrow -\beta
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Frechet1(\alpha,\sigma)
distribution relationship reference: \cite{crooks2010amoroso}

real number
type: Class
subclass of: number

CDF of Logistic 2
type: cumulative density function
has name: CDF of Logistic 2
has Latex expression: \frac{1}{1+e^{-\tau(x-\mu)}}
has R code expression: 1/(1+exp(-tau*(x-mu)))

shape parameter of Kumaraswamy-1
type: mathematical object
has name: shape parameter of Kumaraswamy-1
has code name: Kumaraswamy1.shape1
has Latex symbol: a
parameter has definition expressed in plain text: a > 0
has short name: shape parameter
has short code name: shape1

relationship between Hypergeometric 1 and Normal 1 whereby X \\sim Hypergeometric1N, K, n \\Rightarrow Y\\sim Normal1\\mu,\\sigma \\text{ for large n, but K/N not too small}
type: Limiting
distribution relationship from: Hypergeometric 1
distribution relationship to: Normal
distribution relationship definition in Latex: X \sim Hypergeometric1(N, K, n) \Rightarrow Y\sim Normal1(\mu,\sigma) \text{ for large n, but K/N not too small}
distribution relationship pair expression Latex: Hypergeometric1(N,K,n) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{forbes2011statistical}

lower limit of Triangular-1
type: mathematical object
has name: lower limit of Triangular-1
has code name: Triangular1.lowerLimit
has Latex symbol: a
parameter has definition expressed in plain text: a \in R
has short name: lower limit
has short code name: lowerLimit

CDF of Logit Normal 1
type: cumulative density function
has name: CDF of Logit Normal 1
has Latex expression: \frac12\Big[1 + \operatorname{erf}\Big( \frac{\operatorname{logit}(x)-\mu}{\sqrt{2\sigma^2}}\Big)\Big]
has R code expression: 1/2*(1 + erf((logit(x)-mu)/sqrt(2*sigma^2))))

YuleSimon1
type: probability distribution
has name: YuleSimon1
has code name: YuleSimon1
distribution has PMF: PMF of YuleSimon1
distribution has CDF: CDF of YuleSimon1
distribution has mean: mean of YuleSimon1
distribution has mode: mode of YuleSimon1
distribution has variance: variance of YuleSimon1
has variate symbol in Latex: k
has Latex expression of support: k \in \{1,2,\dots\}
distribution has parameter: shape parameter of YuleSimon1

Hyperbolic secant 1
type: probability distribution
has name: Hyperbolic secant 1
has code name: HyperbolicSecant1
distribution has PDF: PDF of Hyperbolic secant 1
distribution has CDF: CDF of Hyperbolic secant 1
distribution has mean: mean of Hyperbolic secant 1
distribution has median: median of Hyperbolic secant 1
distribution has mode: mode of Hyperbolic secant 1
distribution has variance: variance of Hyperbolic secant 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)

PDF of Folded Normal 1
type: probability density function
has name: PDF of Folded Normal 1
has Latex expression: \frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x-\mu)^2}{2\sigma^2} } + \frac{1}{\sigma\sqrt{2\pi}} \,e^{ -\frac{(x+\mu)^2}{2\sigma^2} }
has R code expression: 1/(sigma*sqrt(2*pi))*exp(-(x-mu)^2/(2*sigma^2)) + 1/(sigma*sqrt(2*pi))*exp(-(x+mu)^2/(2*sigma^2))

has name
type: DatatypeProperty

index parameter of Inverse-Binomial-1
type: mathematical object
has name: index parameter of Inverse-Binomial-1
has code name: InverseBinomial1.index
has Latex symbol: k
parameter has definition expressed in plain text: k \in \left\{0,1,2,\dots\right\}
has short name: index parameter
has short code name: index

relationship between Gamma 1 and Chi-squared 1 whereby k_{ChiSquared1}=2k, \\theta=2
type: Special case
distribution relationship from: Gamma 1
distribution relationship to: Chi-square
distribution relationship definition in Latex: k_{ChiSquared1}=2k, \theta=2
distribution relationship pair expression Latex: Gamma1(k,\theta) \rightarrow ChiSquared1(n)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaChisquareT.pdf}

variance of Multivariate Normal 1
type: mathematical object
has name: variance of Multivariate Normal 1
has Latex expression: \Sigma

variance of Weibull 1
type: mathematical object
has name: variance of Weibull 1
has Latex expression: \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]

location parameter of Amoroso-1
type: mathematical object
has name: location parameter of Amoroso-1
has code name: Amoroso1.location
has Latex symbol: a
parameter has definition expressed in plain text: a \in R
has short name: location parameter
has short code name: location

logit of probability of success of Bernoulli-2
type: mathematical object
has name: logit of probability of success of Bernoulli-2
has code name: Bernoulli2.logitProbability
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R
has short name: logit of probability of success
has short code name: logitProbability

mean of Truncated-Normal-1
type: mathematical object
has name: mean of Truncated-Normal-1
has code name: TruncatedNormal1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: mean

relationship between GeneralizedPoisson2 and Generalized Poisson 1 whereby \\theta = \\mu 1-\\delta
type: Reparameterisation
distribution relationship from: GeneralizedPoisson2
distribution relationship to: Generalized Poisson 1
distribution relationship definition in Latex: \theta = \mu (1-\delta)
distribution relationship pair expression Latex: GeneralizedPoisson2(\mu,\delta) \rightarrow GeneralizedPoisson1(\theta,\delta)
distribution relationship reference: \cite{Yang:2007vn} \\ ProbOnto spec

mode of Kumaraswamy 1
type: mathematical object
has name: mode of Kumaraswamy 1
has Latex expression: \left(\frac{a-1}{ab-1}\right)^{1/a} \text{ for } a\geq 1, b\geq 1, (a,b)\neq (1,1)

scale parameter of Amoroso-1
type: mathematical object
has name: scale parameter of Amoroso-1
has code name: Amoroso1.scale
has Latex symbol: \theta
parameter has definition expressed in plain text: \theta \in R
has short name: scale parameter
has short code name: scale

mean of Logistic 2
type: mathematical object
has name: mean of Logistic 2
has Latex expression: \mu

relationship between Geometric 1 and Negative Binomial 1 whereby \\Sigma X iid
type: Transformation
distribution relationship from: Geometric 1
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: \Sigma X (iid)
distribution relationship pair expression Latex: Geometric1(p) \rightarrow NegativeBinomial1(r,p)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GeometricPascal.pdf}

variance of Standard Logistic 1
type: mathematical object
has name: variance of Standard Logistic 1
has Latex expression: \frac{\pi^2}{3}

mean of Generalized Gamma 2
type: mathematical object
has name: mean of Generalized Gamma 2
has Latex expression: a + b\Gamma(c+1/k)/\Gamma(c)

CDF of Beta 1
type: cumulative density function
has name: CDF of Beta 1
has Latex expression: I_x(\alpha,\beta)
has R code expression: Rbeta(x, alpha, beta)

rational number
type: Class
subclass of: real number

relationship between Kumaraswamy 1 and Exponential 1 whereby \\text{If } X \\sim Kumaraswamy1a,1 \\text{ then } -\\logX \\sim Exponential1a
type: Transformation
distribution relationship from: Minimax distribution
distribution relationship to: Exponential 1
distribution relationship definition in Latex: \text{If } X \sim Kumaraswamy1(a,1) \text{ then } -\log(X) \sim Exponential1(a)
distribution relationship pair expression Latex: Kumaraswamy1(a,b) \rightarrow Exponential1(a)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Kumaraswamy_distribution}

CDF of Power 1
type: cumulative density function
has name: CDF of Power 1
has Latex expression: \frac{x^\beta}{\alpha^\beta}
has R code expression: x^beta / alpha^beta

mean of Triangular 1
type: mathematical object
has name: mean of Triangular 1
has Latex expression: (a+b+c)/3

mean of YuleSimon1
type: mathematical object
has name: mean of YuleSimon1
has Latex expression: \frac{\rho}{\rho-1} \text{ for } \rho>1

mean of Logit Normal 1
type: mathematical object
has name: mean of Logit Normal 1
has Latex expression: \text{ no analytical solution }

variance of Half-normal 1
type: mathematical object
has name: variance of Half-normal 1
has Latex expression: \frac{\pi - 2}{2 \theta^2}

variance of Truncated Normal 1
type: mathematical object
has name: variance of Truncated Normal 1
has Latex expression: \sigma^2 \Big[ 1 + \frac{ \frac{a-\mu}{\sigma}\phi(\frac{a-\mu}{\sigma}) - \frac{b-\mu}{\sigma}\phi(\frac{b-\mu}{\sigma}) }{ \Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma}) } - \Big( \frac{ \phi(\frac{a-\mu}{\sigma}) - \phi(\frac{b-\mu}{\sigma}) }{ \Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma}) } \Big)^2 \Big]

relationship between Truncated Normal 1 and Normal 1 whereby a=-\\infty, b=\\infty
type: Special case
distribution relationship from: Truncated Normal 1
distribution relationship to: Normal
distribution relationship definition in Latex: a=-\infty, b=\infty
distribution relationship pair expression Latex: TruncatedNormal1(\mu,\sigma,a,b) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{forbes2011statistical}

CDF of Trapezoidal 1
type: cumulative density function
has name: CDF of Trapezoidal 1
has Latex expression: \left\{ \begin{array}{rcl} \dfrac{1}{d+c-a-b}\dfrac{1}{b-a}(x-a)^2 & a\leq x < b & \\ \dfrac{1}{d+c-a-b} (2x-a-b) & b\leq x < c \\ 1-\dfrac{1}{d+c-a-b}\dfrac{1}{d-c}(d-x)^2 & c\leq x \leq d \end{array}\right. \nonumber \\
has R code expression: CDF1 = 1/(d+c-a-b)*1/(b-a)*(x-a)^2 CDF2 = 1/(d+c-a-b)*(2*x-a-b) CDF3 = 1-1/(d+c-a-b)*1/(d-c)*(d-x)^2

precision matrix of Multivariate-Normal-2
type: mathematical object
has name: precision matrix of Multivariate-Normal-2
has code name: MultivariateNormal2.precisionMatrix
has Latex symbol: T
parameter has definition expressed in plain text: \text{inverse of the covariance matrix}
has short name: precision matrix
has short code name: precisionMatrix

Gumbel 1
type: probability distribution
has name: Gumbel 1
has code name: Gumbel1
distribution has PDF: PDF of Gumbel 1
distribution has CDF: CDF of Gumbel 1
distribution has mean: mean of Gumbel 1
distribution has median: median of Gumbel 1
distribution has mode: mode of Gumbel 1
distribution has variance: variance of Gumbel 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location of Gumbel-1
distribution has parameter: scale of Gumbel-1

CDF of Multivariate Normal 1
type: cumulative density function
has name: CDF of Multivariate Normal 1
has Latex expression: \text{no analytic expression}

mode of Weibull 1
type: mathematical object
has name: mode of Weibull 1
has Latex expression: \begin{cases} \lambda \left(\frac{k-1}{k} \right)^{\frac{1}{k}}\, &k>1\\ 0 &k=1\end{cases}

mode of Hypergeometric 1
type: mathematical object
has name: mode of Hypergeometric 1
has Latex expression: \left \lfloor \frac{(n+1)(K+1)}{N+2} \right \rfloor

hazard function
type: Class
subclass of: mathematical function

relationship between Negative Binomial 1 and Negative Binomial 2 whereby \\tau = 1/r, \\lambda = r1-p/p
type: Reparameterisation
distribution relationship from: Negative Binomial 1
distribution relationship to: Negative Binomial 2
distribution relationship definition in Latex: \tau = 1/r, \lambda = r(1-p)/p
distribution relationship pair expression Latex: NegativeBinomial1(r, p) \rightarrow NegativeBinomial2(\lambda, \tau)
distribution relationship reference: ProbOnto spec

Folded Normal 1
type: probability distribution
has name: Folded Normal 1
has code name: FoldedNormal1
distribution has PDF: PDF of Folded Normal 1
distribution has CDF: CDF of Folded Normal 1
distribution has mean: mean of Folded Normal 1
distribution has variance: variance of Folded Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: scale of Folded-Normal-1
distribution has parameter: location of Folded-Normal-1

has alternate name
type: DatatypeProperty

mode of Standard Logistic 1
type: mathematical object
has name: mode of Standard Logistic 1
has Latex expression: 0

Poisson intensity of Negative-Binomial-2
type: mathematical object
has name: Poisson intensity of Negative-Binomial-2
has code name: NegativeBinomial2.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in R, \lambda > 0
has short name: Poisson intensity
has short code name: rate

median of Logistic 2
type: mathematical object
has name: median of Logistic 2
has Latex expression: \mu

median of Kumaraswamy 1
type: mathematical object
has name: median of Kumaraswamy 1
has Latex expression: \left(1-2^{-1/b}\right)^{1/a}

relationship between Gamma 2 and Gamma 1 whereby k=r, \\theta = 1 / \\mu
type: Reparameterisation
distribution relationship from: Gamma 2
distribution relationship to: Gamma 1
distribution relationship definition in Latex: k=r, \theta = 1 / \mu
distribution relationship pair expression Latex: Gamma2(r,mu) \rightarrow Gamma1(k,\theta)
distribution relationship reference: ProbOnto spec

relationship between Student's t-distribution 3 and Student's t-distribution 2 whereby \\tau=1/\\sigma^2
type: Reparameterisation
distribution relationship from: non-standardized Student's t-distribution
distribution relationship to: t-distribution
distribution relationship definition in Latex: \tau=1/\sigma^2
distribution relationship pair expression Latex: StudentT3(\nu,\mu,\sigma) \rightarrow StudentT2(\mu,\tau,k)
distribution relationship reference: ProbOnto spec

PROB k0001410
type: Special case
label: relationship between Amoroso 1 and Weibull 1 whereby \\text{Weibull1}\\lambda,k \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a = 0, \\theta=\\lambda, \\alpha=1
distribution relationship from: Amoroso 1
distribution relationship to: Weibull 1
distribution relationship definition in Latex: \text{Weibull1}(\lambda,k) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a = 0, \theta=\lambda, \alpha=1
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Weibull1(\lambda,k)
distribution relationship reference: \cite{crooks2010amoroso}

relationship between Scaled Inverse Chi-Square and Inverse-Gamma 1 whereby \\text{If } X \\sim ScaledInverseChiSquare1\\nu,\\sigma \\text{ then } X \\sim InverseGamma1\\nu/2,\\nu \\sigma^2 /2
type: Transformation
distribution relationship from: Scaled Inverse Chi-Square
distribution relationship to: Inverse-Gamma 1
distribution relationship definition in Latex: \text{If } X \sim ScaledInverseChiSquare1(\nu,\sigma) \text{ then } X \sim InverseGamma1(\nu/2,\nu \sigma^2 /2)
distribution relationship pair expression Latex: ScaledInverseChiSquare1(\nu,\sigma) \rightarrow InverseGamma1(\alpha,\beta)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution}

Generalized Gamma 2
type: probability distribution
has name: Generalized Gamma 2
has code name: GeneralizedGamma2
distribution has PDF: PDF of Generalized Gamma 2
distribution has CDF: CDF of Generalized Gamma 2
distribution has mean: mean of Generalized Gamma 2
distribution has mode: mode of Generalized Gamma 2
distribution has variance: variance of Generalized Gamma 2
has variate symbol in Latex: x
has Latex expression of support: 0 < a < x
distribution has parameter: shape1 of Generalized-Gamma-2
distribution has parameter: scale of Generalized-Gamma-2
distribution has parameter: location of Generalized-Gamma-2
distribution has parameter: shape2 of Generalized-Gamma-2

inverse scale of Half-normal-1
type: mathematical object
has name: inverse scale of Half-normal-1
has code name: HalfNormal1.inverseScale
has Latex symbol: \theta
parameter has definition expressed in plain text: \theta > 0
has short name: inverse scale
has short code name: inverseScale

mean of Power 1
type: mathematical object
has name: mean of Power 1
has Latex expression: \frac{\alpha \beta}{1+\beta}

NA of Makeham-1
type: mathematical object
has name: NA of Makeham-1
has code name: Makeham1.delta
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short code name: delta

CDF of YuleSimon1
type: cumulative density function
has name: CDF of YuleSimon1
has Latex expression: 1 - kB(k, \rho+1)
has R code expression: 1 - k*beta(k,rho+1)

mode of Triangular 1
type: mathematical object
has name: mode of Triangular 1
has Latex expression: c

relationship between Negative Binomial 2 and Negative Binomial 4 whereby r=1/\\tau, p = \\frac{\\tau \\lambda}{1 + \\tau \\lambda}
type: Reparameterisation
distribution relationship from: Negative Binomial 2
distribution relationship to: Negative Binomial 4
distribution relationship definition in Latex: r=1/\tau, p = \frac{\tau \lambda}{1 + \tau \lambda}
distribution relationship pair expression Latex: NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial4(r,p)
distribution relationship reference: ProbOnto spec \\ \cite{cameron2013regression}

mean of Truncated Normal 1
type: mathematical object
has name: mean of Truncated Normal 1
has Latex expression: \mu + \frac{ \phi(\frac{a-\mu}{\sigma}) - \phi(\frac{b-\mu}{\sigma}) }{ \Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma}) } \sigma

mean of Half-normal 1
type: mathematical object
has name: mean of Half-normal 1
has Latex expression: 1/\theta

median of Weibull 1
type: mathematical object
has name: median of Weibull 1
has Latex expression: \lambda(\log(2))^{1/k}

mean of Multivariate Normal 1
type: mathematical object
has name: mean of Multivariate Normal 1
has Latex expression: \mu

shape of Standard-Two-Sided-Power-1
type: mathematical object
has name: shape of Standard-Two-Sided-Power-1
has code name: StandardTwoSidedPower1.shape
has Latex symbol: n
parameter has definition expressed in plain text: n > 0
has short name: shape
has short code name: shape

relationship between Log-Normal 6 and Log-Normal 2 whereby \\mu=\\logm, v=\\log\\sigma_g^2
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 2
distribution relationship definition in Latex: \mu=\log(m), v=\log(\sigma_g^2)
distribution relationship pair expression Latex: LogNormal6(m,\sigma_g) \rightarrow LogNormal2(\mu,v)
distribution relationship reference: ProbOnto spec

scale of Gamma-1
type: mathematical object
has name: scale of Gamma-1
has code name: Gamma1.scale
has Latex symbol: \theta
parameter has definition expressed in plain text: \theta > 0
has short name: scale
has short code name: scale

Amoroso 1
type: probability distribution
has name: Amoroso 1
has code name: Amoroso1
distribution has PDF: PDF of Amoroso 1
has variate symbol in Latex: x
has Latex expression of support: x \geq a \text{ if } \theta > 0, x \leq a \text{ if } \theta < 0
distribution has parameter: scale parameter of Amoroso-1
distribution has parameter: shape paremeter of Amoroso-1
distribution has parameter: shape parameter of Amoroso-1
distribution has parameter: location parameter of Amoroso-1

PDF of Gumbel 1
type: probability density function
has name: PDF of Gumbel 1
has Latex expression: \frac{e^{-e^{-\frac{x-\mu}{\beta}}} e^{-\frac{x-\mu}{\beta}}}{\beta}
has R code expression: (exp(-exp(-(x-mu)/beta)) * exp(-(x-mu)/beta))/beta

survival function
type: Class
subclass of: mathematical function

median of Logit Normal 1
type: mathematical object
has name: median of Logit Normal 1
has Latex expression: \frac{\exp(\mu)}{1+\exp(\mu)}

median of Standard Logistic 1
type: mathematical object
has name: median of Standard Logistic 1
has Latex expression: 0

has abbreviation
type: DatatypeProperty

variance of Log-Normal 6
type: mathematical object
has name: variance of Log-Normal 6
has Latex expression: m^2 e^{\log^2(\sigma_g)} [e^{\log^2(\sigma_g)}-1]

relationship between Kumaraswamy 1 and Standard Uniform 1 whereby \\text{If } X \\sim Kumaraswamy11,1 \\text{ then } X \\sim StandardUniform1
type: Special case
distribution relationship from: Minimax distribution
distribution relationship to: Standard Uniform 1
distribution relationship definition in Latex: \text{If } X \sim Kumaraswamy1(1,1) \text{ then } X \sim StandardUniform1
distribution relationship pair expression Latex: Kumaraswamy1(a,b) \rightarrow StandardUniform1
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Kumaraswamy_distribution}

mean of Kumaraswamy 1
type: mathematical object
has name: mean of Kumaraswamy 1
has Latex expression: \frac{b\Gamma(1+\tfrac{1}{a})\Gamma(b)}{\Gamma(1+\tfrac{1}{a}+b)}

mode of Logistic 2
type: mathematical object
has name: mode of Logistic 2
has Latex expression: \mu

PROB k0001438
type: Transformation
label: relationship between Beta Negative Binomial1 and Negative Binomial 1 whereby \\text{If } X \\sim \\text{NegativeBinomial1n,p and } p \\sim \\text{Beta1}\\alpha,\\beta \\text{ then } X \\sim \\text{NegativeBinomial1}\\alpha,\\beta,n
distribution relationship from: Beta Negative Binomial1
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: \text{If } X \sim \text{NegativeBinomial1(n,p) and } p \sim \text{Beta1(}\alpha,\beta \text{) then } X \sim \text{NegativeBinomial1}(\alpha,\beta,n)
distribution relationship pair expression Latex: BetaNegativeBinomial1(\alpha,\beta,n) \rightarrow NegativeBinomial1(r,p)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalBetapascal.pdf}

relationship between Beta-binomial 1 and Uniform Discrete 2 whereby \\alpha=1, \\beta=1
type: Special case
distribution relationship from: Beta-binomial 1
distribution relationship to: Uniform Discrete 2
distribution relationship definition in Latex: \alpha=1, \beta=1
distribution relationship pair expression Latex: BetaBinomial1(n,\alpha,\beta) \rightarrow UniformDiscrete2(n)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BetabinomialRectangular.pdf}

PROB k0000973
type: Transformation
label: relationship between Conway-Maxwell-Poisson 1 and Negative Binomial 1 whereby \\text{ For } \\nu=0 \\text{ and } \\lambda<1 \\text{ the sum of Conway-Maxwell-Poisson distributed variables}\\\\ \\text{reduces to the sum of geometric variables, which follows a Negative Binomial distribution with parameters}\\\\ n \\text{ and } 1-\\lambda
distribution relationship from: Conway-Maxwell-Poisson 1
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: \text{ For } \nu=0 \text{ and } \lambda<1 \text{ the sum of Conway-Maxwell-Poisson distributed variables}\\ \text{reduces to the sum of geometric variables, which follows a Negative Binomial distribution with parameters}\\ n \text{ and } 1-\lambda
distribution relationship pair expression Latex: ConwayMaxwellPoisson1(\lambda,\nu) \rightarrow NegativeBinomial1(r,p)
distribution relationship reference: \cite{shmueli2005useful}

PDF of Generalized Gamma 2
type: probability density function
has name: PDF of Generalized Gamma 2
has Latex expression: \frac{k (x-a)^{kc-1}}{b^{kc}\Gamma(c)}\exp\Big[-\Big(\frac{x-a}{b}\Big)^k\Big]
has R code expression: (k*(x-a)^(k*c-1)) / (b^(k*c)*gamma(c)) * exp( -((x-a)/b)^k )

variance of Makeham 1
type: mathematical object
has name: variance of Makeham 1
has Latex expression: \text{mathematically intractable}

SF of Power 1
type: survival function
has name: SF of Power 1
has Latex expression: 1- \frac{x^\beta}{\alpha^\beta}
has R code expression: 1- x^beta / alpha^beta

variance of Uniform 1
type: mathematical object
has name: variance of Uniform 1
has Latex expression: \tfrac{1}{12} (b-a)^2

PDF of Muth 1
type: probability density function
has name: PDF of Muth 1
has Latex expression: (e^{\kappa x}-\kappa) e^{\left[-\frac{e^{\kappa x}}{\kappa} + \kappa x + \frac1{\kappa} \right]}
has R code expression: (exp(kappa*x)-kappa)*exp( -exp(kappa*x)/x + kappa*x + 1/kappa )

HF of Arcsine 1
type: hazard function
has name: HF of Arcsine 1
has Latex expression: \frac{2}{\sqrt{x(1-x)} (\pi - 2 \arcsin\left(2x -1\right))}
has R code expression: 2/(sqrt(x*(1-x))*(pi - 2 * asin(2*x -1)))

Power-Normal 1
type: probability distribution
has name: Power-Normal 1
has code name: PowerNormal1
distribution has PDF: PDF of Power-Normal 1
distribution has CDF: CDF of Power-Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: location of Power-Normal-1
distribution has parameter: scale of Power-Normal-1
distribution has parameter: transforming parameter of Power-Normal-1

PDF of Wishart 2
type: probability density function
has name: PDF of Wishart 2
has Latex expression: |R|^{k/2}|x|^{(k-p-1)/2}e^{-\frac{1}2\,tr(Rx)}

PMF of Zero-inflated Poisson 1
type: probability mass function
has name: PMF of Zero-inflated Poisson 1
has Latex expression: \begin{cases} \pi + (1-\pi) e^{-\lambda}& \text{for } k = 0 \\ (1-\pi) e^{-\lambda} \frac{\lambda^k}{k!} & \text{for } k > 0 \end{cases}
has R code expression: PMF1=pi + (1-pi)*exp(-lambda) # if k=0 PMF2=(1-pi)*exp(-lambda) * lambda^k/factorial(k) # if k>0

PDF of Dagum 1
type: probability density function
has name: PDF of Dagum 1
has Latex expression: \frac{a p}{x} \left( \frac{(\tfrac{x}{b})^{a p}}{\left((\tfrac{x}{b})^a + 1 \right)^{p+1}} \right)
has R code expression: a*p/x * (x/b)^(a*p) / ((x/b)^a + 1)^(p+1)

scale of Cauchy-1
type: mathematical object
has name: scale of Cauchy-1
has code name: Cauchy1.scale
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma \in R
has short name: scale
has short code name: scale

relationship between Log-Normal 4 and Log-Normal 2 whereby \\mu = \\logm, v = \\logcv^2+1
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 2
distribution relationship definition in Latex: \mu = \log(m), v = \log(cv^2+1)
distribution relationship pair expression Latex: LogNormal4(m,cv) \rightarrow LogNormal2(\mu,v)
distribution relationship reference: ProbOnto spec

Uniform Discrete 2
type: probability distribution
label: Rectangular Discrete 2
has name: Uniform Discrete 2
has code name: UniformDiscrete2
distribution has PMF: PMF of Uniform Discrete 2
distribution has CDF: CDF of Uniform Discrete 2
distribution has SF: SF of Uniform Discrete 2
distribution has HF: HF of Uniform Discrete 2
distribution has mean: mean of Uniform Discrete 2
distribution has variance: variance of Uniform Discrete 2
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,\dots,n\}
distribution has parameter: number of values of Uniform-Discrete-2

scale parameter of Johnson-SB-1
type: mathematical object
has name: scale parameter of Johnson-SB-1
has code name: JohnsonSB1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: scale parameter
has short code name: scale

mean of Noncentral F 1
type: mathematical object
has name: mean of Noncentral F 1
has Latex expression: \frac{n}{m} \frac{m+lambda}{n-2}

PMF of Poisson 1
type: probability mass function
has name: PMF of Poisson 1
has Latex expression: \frac{\lambda^k}{k!}e^{-\lambda}
has R code expression: lambda^k/factorial(k) * exp(-lambda)

mean of Von Mises 1
type: mathematical object
has name: mean of Von Mises 1
has Latex expression: \mu

Birnbaum-Saunders 1
type: probability distribution
has name: Birnbaum-Saunders 1
has code name: BirnbaumSaunders1
distribution has PDF: PDF of Birnbaum-Saunders 1
distribution has CDF: CDF of Birnbaum-Saunders 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Birnbaum-Saunders-1
distribution has parameter: scale of Birnbaum-Saunders-1

median of Gompertz 1
type: mathematical object
has name: median of Gompertz 1
has Latex expression: \left(1/b\right)\log\left[\left(-1/\eta\right) \log\left(1/2\right)+1\right]

relationship between Log-Normal 5 and Log-Normal 1 whereby \\mu = \\mu, \\sigma = 1 / \\sqrt{\\tau}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \mu = \mu, \sigma = 1 / \sqrt{\tau}
distribution relationship pair expression Latex: LogNormal5(\mu,\tau) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

CDF of Poisson 2
type: cumulative density function
has name: CDF of Poisson 2
has Latex expression: \frac{\gamma(\lfloor k+1 \rfloor,\exp(\alpha))}{\lfloor k \rfloor!}
has R code expression: Igamma(floor(k+1), exp(alpha), lower=F) / factorial(floor(k))

mean of Inverse Binomial 1
type: mathematical object
has name: mean of Inverse Binomial 1
has Latex expression: k (1-p) / (2p-1)

Logit Normal 1
type: probability distribution
has name: Logit Normal 1
has code name: LogitNormal1
distribution has PDF: PDF of Logit Normal 1
distribution has CDF: CDF of Logit Normal 1
distribution has mean: mean of Logit Normal 1
distribution has median: median of Logit Normal 1
distribution has mode: mode of Logit Normal 1
distribution has variance: variance of Logit Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,1)
distribution has parameter: scale of Logit-Normal-1
distribution has parameter: location of Logit-Normal-1

location of Cauchy-1
type: mathematical object
has name: location of Cauchy-1
has code name: Cauchy1.location
has Latex symbol: x_0
parameter has definition expressed in plain text: x_0 \in R
has short name: location
has short code name: location

PDF of Sinh-Arcsinh 1
type: probability density function
has name: PDF of Sinh-Arcsinh 1
has Latex expression: \frac{c}{\sqrt{2\pi} \sigma (1+z^2)^{1/2}} e^{-r^2/2}\\ r = \frac12 \left(\exp[\tau \sinh^{-1}(z)] - \exp[-\nu \sinh^{-1}(z)]\right)\\ c = \frac12 \left( \tau \exp[\tau \sinh^{-1}(z)] + \nu \exp[-\nu \sinh^{-1}(z)] \right)\\ z = (x-\mu)/\sigma
has R code expression: cfunc(x,mu,sigma,nu,tau) / (sqrt(2*pi) * sigma * (1+zfunc(x,mu,sigma)^2)^(1/2)) * exp(-rfunc(x,mu,sigma,nu,tau)^2/2) rfunc = function(x,mu,sigma,nu,tau) { 1/2 * ( exp(tau * asinh(zfunc(x,mu,sigma))) - exp(-nu * asinh(zfunc(x,mu,sigma))) ) } cfunc = function(x,mu,sigma,nu,tau) { 1/2 * ( tau * exp(tau * asinh(zfunc(x,mu,sigma))) + nu * exp(-nu * asinh(zfunc(x,mu,sigma))) ) } zfunc = function(x,mu,sigma) { (x-mu)/sigma }

relationship between Half-normal 2 and Half-normal 1 whereby \\mu=0, \\sigma=\\sqrt{\\pi} / \\theta \\sqrt{2}
type: Reparameterisation
distribution relationship from: Half-normal 2
distribution relationship to: Half-normal 1
distribution relationship definition in Latex: \mu=0, \sigma=\sqrt{\pi} / (\theta \sqrt{2})
distribution relationship pair expression Latex: HalfNormal2(\mu,\sigma) \rightarrow HalfNormal1(\theta)
distribution relationship reference: ProbOnto spec

mean of Categorical Ordered 1
type: mathematical object
has name: mean of Categorical Ordered 1
has Latex expression: E([x=i]) = p_i, \text{this is the mean of the Iverson bracket } [x=i] \text{ and not the mean of } x

mean of Uniform Discrete 1
type: mathematical object
has name: mean of Uniform Discrete 1
has Latex expression: (a+b)/2

scale of Pareto-Type-II
type: mathematical object
has name: scale of Pareto-Type-II
has code name: ParetoTypeII1.scale
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in R^+
has short name: scale
has short code name: scale

CDF of Arcsine 1
type: cumulative density function
has name: CDF of Arcsine 1
has Latex expression: \frac{2}{\pi}\arcsin\left(\sqrt x \right)
has R code expression: 2/pi*arcsin(sqrt(x))

Erlang 1
type: probability distribution
has name: Erlang 1
has code name: Erlang1
distribution has PDF: PDF of Erlang 1
distribution has CDF: CDF of Erlang 1
distribution has SF: SF of Erlang 1
distribution has HF: HF of Erlang 1
distribution has mean: mean of Erlang 1
distribution has mode: mode of Erlang 1
distribution has variance: variance of Erlang 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty) \text{ (Forbes)}, x \in (0,+\infty) \text{ (Leemis)}
distribution has parameter: scale of Erlang-1
distribution has parameter: shape of Erlang-1

Zero-inflated Poisson 1
type: probability distribution
has name: Zero-inflated Poisson 1
has code name: ZeroInflatedPoisson1
distribution has PMF: PMF of Zero-inflated Poisson 1
distribution has CDF: CDF of Zero-inflated Poisson 1
distribution has mean: mean of Zero-inflated Poisson 1
distribution has variance: variance of Zero-inflated Poisson 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: probability of extra zeros of Zero-inflated-Poisson-1
distribution has parameter: Poisson intensity of Zero-inflated-Poisson-1

Poisson 1
type: probability distribution
has name: Poisson 1
has code name: Poisson1
distribution has PMF: PMF of Poisson 1
distribution has CDF: CDF of Poisson 1
distribution has mean: mean of Poisson 1
distribution has median: median of Poisson 1
distribution has mode: mode of Poisson 1
distribution has variance: variance of Poisson 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: Poisson intensity of Poisson-1

PDF of Power-Normal 1
type: probability density function
has name: PDF of Power-Normal 1
has Latex expression: \frac{1}K \frac{1}{\sqrt{2 \pi} \sigma} x^{\lambda-1} \exp\left[-\frac12 \left(\frac{x^{(\lambda)}-\mu}{\sigma}\right)^2 \right] \text{ with } x^{(\lambda)}=\begin{cases} (x^\lambda - 1)/\lambda & \text{for } \lambda \neq 0 \\ \log(x) & \text{for } \lambda = 0 \end{cases}; \quad K=\begin{cases} \Phi(T) & \text{for } \lambda > 0 \\ 1 & \text{for } \lambda = 0 \\ \Phi(-T) & \text{for } \lambda < 0 \end{cases}; \quad T = 1/(\lambda \sigma) + \mu/\sigma
has R code expression: 1/(K(lambda,mu,sigma) * sqrt(2*pi) * sigma) * x^(lambda-1) * exp(- 1/2 * ((BCtrafo(x,lambda)-mu) / sigma)^2 ) BCtrafo = function(x,lambda) { (x^lambda - 1) / lambda } erf <- function(x) { 2 * pnorm(x * sqrt(2)) - 1 } PHI <- function(x) { 1/2 * (1 + erf(x/(sqrt(2)))) } Tfunc <- function(lambda,mu,sigma) { 1/(lambda*sigma) + mu/sigma } K = function(lambda,mu,sigma) { if (lambda > 0) { K = PHI(Tfunc(lambda,mu,sigma)) } else if (lambda==0) { K = 1 } else { K = PHI(-Tfunc(lambda,mu,sigma)) } }

location parameter of Johnson-SB-1
type: mathematical object
has name: location parameter of Johnson-SB-1
has code name: JohnsonSB1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location parameter
has short code name: location

relationship between Generalized Poisson 1 and GeneralizedPoisson2 whereby \\mu = \\theta / 1-\\delta
type: Reparameterisation
distribution relationship from: Generalized Poisson 1
distribution relationship to: GeneralizedPoisson2
distribution relationship definition in Latex: \mu = \theta / (1-\delta)
distribution relationship pair expression Latex: GeneralizedPoisson1(\theta,\delta) \rightarrow GeneralizedPoisson2(\mu,\delta)
distribution relationship reference: \cite{Yang:2007vn} \\ ProbOnto spec

CDF of Noncentral F 1
type: cumulative density function
has name: CDF of Noncentral F 1
has Latex expression: \int_0^x f(x), \text { with f the PDF}
has R code expression: cumsum(PDF*rep(stepSize,length(PDF)))

PDF of Student's t-distribution 2
type: probability density function
has name: PDF of Student's t-distribution 2
has Latex expression: \frac{\Gamma \left(\frac{k+1}{2} \right)} {\Gamma \left(\frac{k}{2} \right)} \sqrt{\frac{\tau}{k\pi}}\left[1+\frac{\tau}{k}(x-\mu)^2 \right]^{-\frac{k+1}{2}}
has R code expression: gamma((k+1)/2)/gamma(k/2)*sqrt(tau/(k*pi))*(1+tau/k*(x-mu)^2)^(-(k+1)/2)

Double Exponential 2
type: probability distribution
label: Laplace 2
has name: Laplace 2
has code name: Laplace2
distribution has PDF: PDF of Laplace 2
distribution has CDF: CDF of Laplace 2
distribution has mean: mean of Laplace 2
distribution has variance: variance of Laplace 2
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location of Laplace-2
distribution has parameter: precision of Laplace-2

CDF of Muth 1
type: cumulative density function
has name: CDF of Muth 1
has Latex expression: 1-e^{\left[-\frac{e^{\kappa x}}{\kappa} + \kappa x + \frac1{\kappa} \right]}
has R code expression: 1-exp( -exp(kappa*x)/x + kappa*x + 1/kappa )

PDF of Birnbaum-Saunders 1
type: probability density function
has name: PDF of Birnbaum-Saunders 1
has Latex expression: \frac{1}{\sqrt{2\pi}}\exp\Big[-\frac{(\sqrt{x/\beta}-\sqrt{\beta/x})^2}{2\gamma^2}\Big]\Big[\frac{\sqrt{x/\beta}+\sqrt{\beta/x}}{2\gamma x}\Big]
has R code expression: 1/(sqrt(2*pi))* exp( -(sqrt(x/beta) - sqrt(beta/x))^2 / (2*gamma^2) ) * ( sqrt(x/beta) + sqrt(beta/x) ) / (2*gamma*x)

CDF of Von Mises 1
type: cumulative density function
has name: CDF of Von Mises 1
has Latex expression: \frac{1}{2\pi I_0(\kappa)}\Big[ x I_0(\kappa) + 2\sum_{j=0}^{\infty}I_j(\kappa)\frac{sin(j(x-mu))}{j}\Big]
has R code expression: 1/(2*pi*besselI(kappa,0,expon.scaled=FALSE)) * ( x*besselI(kappa, 0, expon.scaled = FALSE) + 2*sumFromTo(x,mu,kappa,0,infty)) # with sumFromTo = function(x,mu,kappa,fromValue,toValue) { sigma=0; for(i in fromValue:toValue) { sigma = sigma + besselI(kappa,i,expon.scaled=FALSE) * sin(i*(x-mu))/i } return(sigma) }

concentration of Dirichlet-1
type: mathematical object
has name: concentration of Dirichlet-1
has code name: Dirichlet1.concentration
has Latex symbol: \alpha_1, \cdots, \alpha_K
parameter has definition expressed in plain text: \alpha_1, \cdots, \alpha_K, \alpha_i > 0
has short name: concentration
has short code name: concentration

SF of Gompertz 1
type: survival function
has name: SF of Gompertz 1
has Latex expression: \exp\left(-\eta\left(e^{bx}-1 \right)\right)
has R code expression: exp(-eta*(exp(b*x)-1))

relationship between Log-Normal 4 and Log-Normal 6 whereby m = m, \\sigma_g=\\exp\\!\\big\\sqrt{\\logcv^2+1}\\big
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m = m, \sigma_g=\exp\!\big(\sqrt{\log(cv^2+1)}\big)
distribution relationship pair expression Latex: LogNormal4(m,cv) \rightarrow LogNormal6(m,\sigma_g)
distribution relationship reference: ProbOnto spec

location of Sinh-Arcsinh-1
type: mathematical object
has name: location of Sinh-Arcsinh-1
has code name: SinhArcsinh1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

relationship between Trapezoidal 1 and Triangular 1 whereby b=c
type: Special case
distribution relationship from: Trapezoidal 1
distribution relationship to: Triangular 1
distribution relationship definition in Latex: b=c
distribution relationship pair expression Latex: Trapezoidal1(a,b,c,d) \rightarrow Triangular1(a,b,c)
distribution relationship reference: \cite{Garvey:2000yq}

PMF of Poisson 2
type: probability mass function
has name: PMF of Poisson 2
has Latex expression: \frac1{k!}\exp\big(k \alpha - \exp(\alpha)\big)
has R code expression: 1/factorial(k) * exp(k*alpha - exp(alpha))

variance of Inverse Binomial 1
type: mathematical object
has name: variance of Inverse Binomial 1
has Latex expression: k p (1-p) / (2p - 1)^3

CDF of Categorical Ordered 1
type: cumulative density function
has name: CDF of Categorical Ordered 1
has Latex expression: \begin{cases} 0 & \text{for }x<1 \\ \sum_{j=1}^i p_j & \text{for }x \in [i,i+1) \\ 1 & \text{for }x \geq k \end{cases}

tail index of Pareto-Type-II
type: mathematical object
has name: tail index of Pareto-Type-II
has code name: ParetoTypeII1.tailIndex
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0, \alpha \in R
has short name: tail index
has short code name: tailIndex

CDF of Uniform Discrete 2
type: cumulative density function
has name: CDF of Uniform Discrete 2
has Latex expression: \frac{k+1}{n+1}
has R code expression: (k+1)/(n+1)

relationship between Zero-Inflated Negative Binomial 1 and Zero-inflated Poisson 1 whereby \\tau \\rightarrow 0
type: Limiting
distribution relationship from: Zero-Inflated Negative Binomial 1
distribution relationship to: Zero-inflated Poisson 1
distribution relationship definition in Latex: \tau \rightarrow 0
distribution relationship pair expression Latex: ZeroInflatedNegativeBinomial1(\lambda,\tau,p0) \rightarrow ZeroInflatedPoisson1(\lambda,\pi)
distribution relationship reference: \cite{Troconiz:2009fv}

PDF of Chi-squared 1
type: probability density function
has name: PDF of Chi-squared 1
has Latex expression: \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}\; x^{\frac{k}{2}-1} e^{-\frac{x}{2}}
has R code expression: 1/( 2^k/2 * gamma(k/2) ) * x^(k/2-1) * exp(-x/2)

overdispersion of Negative-Binomial-2
type: mathematical object
has name: overdispersion of Negative-Binomial-2
has code name: NegativeBinomial2.overdispersion
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau \in R
has short name: overdispersion
has short code name: overdispersion

inverse scale of Negative-Binomial-5
type: mathematical object
has name: inverse scale of Negative-Binomial-5
has code name: NegativeBinomial5.inverseScale
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta \in R^+
has short name: inverse scale
has short code name: inverseScale

relationship between Generalized Negative Binomial 1 and Negative Binomial 4 whereby \\beta=1 \\text{ and set } m=r, \\theta=p
type: Reparameterisation
distribution relationship from: Generalized Negative Binomial 1
distribution relationship to: Negative Binomial 4
distribution relationship definition in Latex: \beta=1 \text{ and set } m=r, \theta=p
distribution relationship pair expression Latex: GeneralizedNegativeBinomial1(\theta,\beta,m) \rightarrow NegativeBinomial4(r,p)
distribution relationship reference: \cite{Consul:2006qf}

location of Gumbel-1
type: mathematical object
has name: location of Gumbel-1
has code name: Gumbel1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

relationship between Generalized Gamma 2 and Chi-squared 1 whereby a=0, b=2, c=k_{ChiSquare1}/2,k=1
type: Reparameterisation
distribution relationship from: Generalized Gamma 2
distribution relationship to: Chi-square
distribution relationship definition in Latex: a=0, b=2, c=k_{ChiSquare1}/2,k=1
distribution relationship pair expression Latex: GeneralizedGamma2(a,b,c,k) \rightarrow ChiSquared1(k)
distribution relationship reference: \cite{forbes2011statistical}

PDF of Von Mises 1
type: probability density function
has name: PDF of Von Mises 1
has Latex expression: \frac{\exp(\kappa \cos(x-\mu))}{2\pi I_0(\kappa)}
has R code expression: exp(kappa * cos(x-mu)) / (2*pi * besselI(kappa, 0, expon.scaled = FALSE))

relationship between Log-Logistic 2 and Log-Logistic 1 whereby \\alpha=1/\\lambda, \\beta=\\kappa
type: Reparameterisation
distribution relationship from: Fisk
distribution relationship to: Log-Logistic 1
distribution relationship definition in Latex: \alpha=1/\lambda, \beta=\kappa
distribution relationship pair expression Latex: LogLogistic2(\lambda,\kappa) \rightarrow LogLogistic1(\alpha,\beta)
distribution relationship reference: ProbOnto spec

shape parameter of Johnson-SB-1
type: mathematical object
has name: shape parameter of Johnson-SB-1
has code name: JohnsonSB1.shape2
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short name: shape parameter
has short code name: shape2

HF of Muth 1
type: hazard function
has name: HF of Muth 1
has Latex expression: e^{\kappa x} - \kappa
has R code expression: exp(kappa*x) - kappa

dispersion of Zero-Inflated-Generalized-Poisson-1
type: mathematical object
has name: dispersion of Zero-Inflated-Generalized-Poisson-1
has code name: ZeroInflatedGeneralizedPoisson1.dispersion
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > -1, \alpha \in R
has short name: dispersion
has short code name: dispersion

mean of Poisson 1
type: mathematical object
has name: mean of Poisson 1
has Latex expression: \lambda

PDF of Noncentral F 1
type: probability density function
has name: PDF of Noncentral F 1
has Latex expression: e^{-\frac\lambda2} \sum_{r=0}^{\infty}\frac1{r!}\left(\frac{\lambda}{2}\right)^r \frac{\Gamma\left(\frac{m+n}{2}+r\right)}{\Gamma\left(\frac{m}{2}+r\right)\Gamma\left(\frac{n}{2}\right)} \left(\frac{m}{n}\right)^{\frac{m}{2}+r} \frac{(F')^{\frac{m}{2}-1+r}}{\left(1+\frac{mF'}{n}\right)^{\frac12 (m+n)+r}}
has R code expression: PDF = function(x,lambda,m,n,lowerLimit,upperLimit) { exp(-lambda/2) * PDFsum(x,lambda,m,n,lowerLimit,upperLimit) } PDFsum = function(x,lambda,m,n,fromValue,toValue) { PDF=array(0,length(x)); for(j in 1:length(x)) { for(r in fromValue:toValue) { PDF[j] = PDF[j] + 1/factorial(r)*(lambda/2)^r*gamma((m+n)/2+r)/gamma(m/2+r)/gamma(n/2) * (m/n)^(m/2+r) * x[j]^(m/2-1+r) / (1+m*x[j]/n)^(1/2*(m+n)+r) } } return(PDF) }

mode of Student's t-distribution 2
type: mathematical object
has name: mode of Student's t-distribution 2
has Latex expression: \mu \text{ for } k > 1

relationship between Log-Normal 3 and Log-Normal 4 whereby m = m, cv = \\sqrt{\\exp\\sigma^2 - 1}
type: Reparameterisation
distribution relationship from: Log-Normal 3
distribution relationship to: lognormal
distribution relationship definition in Latex: m = m, cv = \sqrt{\exp(\sigma^2) - 1}
distribution relationship pair expression Latex: LogNormal3(m,\sigma) \rightarrow LogNormal4(m,cv)
distribution relationship reference: ProbOnto spec

PMF of Inverse Binomial 1
type: probability mass function
has name: PMF of Inverse Binomial 1
has Latex expression: \frac{k \; \Gamma(2x + k)}{\Gamma(x+1) \;\Gamma(x + k + 1)} \; p^{k+x} (1-p)^x
has R code expression: (k * gamma(2*x+k)) / (gamma(x+1) * gamma(x+k+1)) * p^(x+k) * (1-p)^x

variance of Negative Binomial 3
type: mathematical object
has name: variance of Negative Binomial 3
has Latex expression: \mu + \mu^2 / \phi

HF of Gompertz 1
type: hazard function
has name: HF of Gompertz 1
has Latex expression: b\eta\exp(-b x)
has R code expression: b *eta * exp(-b*x)

variance of Dirichlet 1
type: mathematical object
has name: variance of Dirichlet 1
has Latex expression: Var[X_i] = \frac{\alpha_i (\alpha_0-\alpha_i)}{\alpha_0^2 (\alpha_0+1)} \quad \text{where} \quad \alpha_0 = \sum_{i=1}^K\alpha_i; \quad Cov[X_i,X_j] = \frac{- \alpha_i \alpha_j}{\alpha_0^2 (\alpha_0+1)}~~(i\neq j)

location of Multivariate-Student-T-2
type: mathematical object
has name: location of Multivariate-Student-T-2
has code name: MultivariateStudentT2.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu = [\mu_1, \dots, \mu_d]^T, \mu_i \in R
has short name: location
has short code name: mean

PROB k0000943
type: Limiting
label: relationship between F 1 and Chi-squared 1 whereby \\text{If } X \\sim F1n_1,n_2 \\text{, the limiting distribution of } n_1X \\text{ as } n_2 \\rightarrow \\infty \\text{ is} \\\\ \\text{ the chi-square distribution with } n_1 \\text{ degrees of freedom}
distribution relationship from: F 1
distribution relationship to: Chi-square
distribution relationship definition in Latex: \text{If } X \sim F1(n_1,n_2) \text{, the limiting distribution of } n_1X \text{ as } n_2 \rightarrow \infty \text{ is} \\ \text{ the chi-square distribution with } n_1 \text{ degrees of freedom}
distribution relationship pair expression Latex: F1(n_1,n_2) \rightarrow ChiSquared1(n)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/FChisquare.pdf}

variance of Uniform Discrete 1
type: mathematical object
has name: variance of Uniform Discrete 1
has Latex expression: \frac{(b-a+1)^2 -1}{12}

PMF of Categorical Ordered 1
type: probability mass function
has name: PMF of Categorical Ordered 1
has Latex expression: p(x=i)=p_i

degrees of freedom of Inverse-Chi-Square
type: mathematical object
has name: degrees of freedom of Inverse-Chi-Square
has code name: InverseChiSquare1.degreesOfFreedom
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu \in R^+
has short name: degrees of freedom
has short code name: degreesOfFreedom

Wishart 2
type: probability distribution
has name: Wishart 2
has code name: Wishart2
distribution has PDF: PDF of Wishart 2
has variate symbol in Latex: X
has Latex expression of support: X(p \times p) - \text{symmetric, positive definite matrix}
distribution has parameter: inverse scale matrix of Wishart-2
distribution has parameter: degrees of freedom of Wishart-2

shape of Negative-Binomial-5
type: mathematical object
has name: shape of Negative-Binomial-5
has code name: NegativeBinomial5.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R^+
has short name: shape
has short code name: shape

Von Mises 1
type: probability distribution
has name: Von Mises 1
has code name: VonMises1
distribution has PDF: PDF of Von Mises 1
distribution has CDF: CDF of Von Mises 1
distribution has mean: mean of Von Mises 1
distribution has median: median of Von Mises 1
distribution has mode: mode of Von Mises 1
distribution has variance: variance of Von Mises 1
has variate symbol in Latex: x
has Latex expression of support: x \in \text{any interval of length } 2\pi
distribution has parameter: concentration of Von-Mises-1
distribution has parameter: location of Von-Mises-1

relationship between Inverse Gaussian 1 and Chi-squared 1 whereby X \\sim InverseGaussian1\\lambda,\\mu \\text{ and } Y = \\lambda X-\\mu^2 / \\mu^2 X \\Rightarrow Y \\sim ChiSquared1k
type: Transformation
distribution relationship from: Inverse Gaussian 1
distribution relationship to: Chi-square
distribution relationship definition in Latex: X \sim InverseGaussian1(\lambda,\mu) \text{ and } Y = \lambda (X-\mu)^2 / (\mu^2 X) \Rightarrow Y \sim ChiSquared1(k)
distribution relationship pair expression Latex: InverseGaussian1(\lambda,\mu) \rightarrow ChiSquared1(k)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/InversegaussianChisquare.pdf}

Noncentral F 1
type: probability distribution
has name: Noncentral F 1
has code name: NoncentralF1
distribution has PDF: PDF of Noncentral F 1
distribution has CDF: CDF of Noncentral F 1
distribution has mean: mean of Noncentral F 1
distribution has variance: variance of Noncentral F 1
has variate symbol in Latex: F'=\frac{x_1/m}{x_2/n} \text{ with } x_1 \sim ChiSquared1(m), x_2 \sim ChiSquared1(n)
has Latex expression of support: F' \in (0,+\infty)
distribution has parameter: degrees of freedom of Noncentral-F-1
distribution has parameter: noncentrality parameter of Noncentral-F-1

variance of Gumbel 1
type: mathematical object
has name: variance of Gumbel 1
has Latex expression: \frac{\pi^2}{6} \beta^2

relationship between Negative Binomial 3 and Negative Binomial 5 whereby \\alpha = \\phi, \\beta = \\phi / \\mu
type: Reparameterisation
distribution relationship from: Negative Binomial 3
distribution relationship to: Negative Binomial 5
distribution relationship definition in Latex: \alpha = \phi, \beta = \phi / \mu
distribution relationship pair expression Latex: NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial5(\alpha, \beta)
distribution relationship reference: ProbOnto spec

probability of zero of Zero-Inflated-Generalized-Poisson-1
type: mathematical object
has name: probability of zero of Zero-Inflated-Generalized-Poisson-1
has code name: ZeroInflatedGeneralizedPoisson1.probabilityOfZero
has Latex symbol: p0
parameter has definition expressed in plain text: 0<p0<1, p0 \in R
has short name: probability of zero
has short code name: probabilityOfZero

shape parameter of Johnson-SB-1
type: mathematical object
has name: shape parameter of Johnson-SB-1
has code name: JohnsonSB1.shape1
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma \in R
has short name: shape parameter
has short code name: shape1

shape parameter of Johnson-SN-1
type: mathematical object
has name: shape parameter of Johnson-SN-1
has code name: JohnsonSN1.shape1
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma \in R
has short name: shape parameter
has short code name: shape1

relationship between Poisson 1 and Poisson 2 whereby \\alpha = \\log\\lambda
type: Transformation
distribution relationship from: Poisson 1
distribution relationship to: Poisson 2
distribution relationship definition in Latex: \alpha = \log(\lambda)
distribution relationship pair expression Latex: Poisson1(\lambda) \rightarrow Poisson2(\alpha)
distribution relationship reference: \cite{stan-manual:2015b}

SF of Muth 1
type: survival function
has name: SF of Muth 1
has Latex expression: e^{\left[-\frac{e^{\kappa x}}{\kappa} + \kappa x + \frac1{\kappa} \right]}
has R code expression: exp( -exp(kappa*x)/x + kappa*x + 1/kappa )

mean of Student's t-distribution 2
type: mathematical object
has name: mean of Student's t-distribution 2
has Latex expression: \mu

CDF of Poisson 1
type: cumulative density function
has name: CDF of Poisson 1
has Latex expression: \frac{\gamma(\lfloor k+1 \rfloor,\lambda)}{\lfloor k \rfloor!}
has R code expression: Igamma(floor(k+1), lambda, lower=F) / factorial(floor(k))

PMF of Uniform Discrete 2
type: probability mass function
has name: PMF of Uniform Discrete 2
has Latex expression: 1/(n+1)
has R code expression: 1/(n+1)

variance of Cauchy 1
type: mathematical object
has name: variance of Cauchy 1
has Latex expression: undefined

CDF of Gompertz 1
type: cumulative density function
has name: CDF of Gompertz 1
has Latex expression: 1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)
has R code expression: 1-exp(-eta*(exp(b*x)-1))

relationship between Negative Binomial 6 and Negative Binomial 3 whereby \\mu = \\exp\\eta
type: Reparameterisation
distribution relationship from: Negative Binomial 6
distribution relationship to: Negative Binomial 3
distribution relationship definition in Latex: \mu = \exp(\eta)
distribution relationship pair expression Latex: NegativeBinomial6(\eta,\phi) \rightarrow NegativeBinomial3(\mu,\phi)
distribution relationship reference: \cite{stan-manual:2015b}

scale parameter of Johnson-SL-1
type: mathematical object
has name: scale parameter of Johnson-SL-1
has code name: JohnsonSL1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: scale parameter
has short code name: scale

relationship between GeneralizedPoisson2 and Poisson 1 whereby \\delta = 0, \\lambda=\\mu
type: Special case
distribution relationship from: GeneralizedPoisson2
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \delta = 0, \lambda=\mu
distribution relationship pair expression Latex: GeneralizedPoisson2(\mu,\delta) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{Plan:2014kq}

median of Uniform Discrete 1
type: mathematical object
has name: median of Uniform Discrete 1
has Latex expression: (a+b)/2

relationship between Normal 2 and Normal 1 whereby \\mu = \\mu, \\sigma = \\sqrt{v}
type: Reparameterisation
distribution relationship from: Normal
distribution relationship to: Normal
distribution relationship definition in Latex: \mu = \mu, \sigma = \sqrt{v}
distribution relationship pair expression Latex: Normal2(\mu,v) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

Categorical Ordered 1
type: probability distribution
has name: Categorical Ordered 1
has code name: CategoricalOrdered1
distribution has PMF: PMF of Categorical Ordered 1
distribution has CDF: CDF of Categorical Ordered 1
distribution has mean: mean of Categorical Ordered 1
distribution has median: median of Categorical Ordered 1
distribution has mode: mode of Categorical Ordered 1
distribution has variance: variance of Categorical Ordered 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{1,\dots,k\}
distribution has parameter: category probabilities of Categorical-Ordered-1

Sinh-Arcsinh 1
type: probability distribution
label: SHASH
has name: Sinh-Arcsinh 1
has code name: SinhArcsinh1
distribution has PDF: PDF of Sinh-Arcsinh 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: left hand tail of Sinh-Arcsinh-1
distribution has parameter: location of Sinh-Arcsinh-1
distribution has parameter: right hand tail of Sinh-Arcsinh-1
distribution has parameter: scale of Sinh-Arcsinh-1

Binomial 1
type: probability distribution
has name: Binomial 1
has code name: Binomial1
distribution has PMF: PMF of Binomial 1
distribution has CDF: CDF of Binomial 1
distribution has mean: mean of Binomial 1
distribution has median: median of Binomial 1
distribution has mode: mode of Binomial 1
distribution has variance: variance of Binomial 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,\dots,n\}
distribution has parameter: success probability in each trial of Binomial-1
distribution has parameter: number of trials of Binomial-1

shape parameter of Johnson-SN-1
type: mathematical object
has name: shape parameter of Johnson-SN-1
has code name: JohnsonSN1.shape2
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short name: shape parameter
has short code name: shape2

mean of Muth 1
type: mathematical object
has name: mean of Muth 1
has Latex expression: 1

CDF of Johnson SB 1
type: cumulative density function
has name: CDF of Johnson SB 1
has Latex expression: \begin{cases} \frac12 \text{erfc}\left[-\frac{\gamma + \delta \,\log\left[\frac{x-\mu}{-x+\mu+\sigma}\right]}{\sqrt{2}}\right] & \text{for } \mu < x < \mu + \frac{\sigma}{2} \\ \frac12\left(1+ \text{erf}\left[\frac{\gamma + \delta \,\log\left[\frac{x-\mu}{-x+\mu+\sigma}\right]}{\sqrt{2}}\right] \right) & \text{for }\mu + \frac{\sigma}{2} \le x < \mu + \sigma \end{cases}
has R code expression: CDF1=1/2*erfc(-(gamma+delta*log((x-mu)/(-x+mu+sigma)))/sqrt(2)) CDF2=1/2*(1+erf((gamma+delta*log((x-mu)/(-x+mu+sigma)))/sqrt(2)))

mode of Poisson 1
type: mathematical object
has name: mode of Poisson 1
has Latex expression: \lceil\lambda\rceil - 1, \lfloor\lambda\rfloor

probability of extra zeros of Zero-inflated-Poisson-1
type: mathematical object
has name: probability of extra zeros of Zero-inflated-Poisson-1
has code name: ZeroInflatedPoisson1.probabilityOfZero
has Latex symbol: \pi
parameter has definition expressed in plain text: 0<\pi<1, \pi \in R
has short name: probability of extra zeros
has short code name: probabilityOfZero

location of Power-Normal-1
type: mathematical object
has name: location of Power-Normal-1
has code name: PowerNormal1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

variance of Negative Binomial 5
type: mathematical object
has name: variance of Negative Binomial 5
has Latex expression: \alpha/\beta^2 (\beta+1)

SF of Uniform Discrete 2
type: survival function
has name: SF of Uniform Discrete 2
has Latex expression: \frac{n-k}{n+1}
has R code expression: (n-k)/(n+1)

mode of Inverse Chi-Square
type: mathematical object
has name: mode of Inverse Chi-Square
has Latex expression: \frac{1}{\nu + 2}

relationship between Standard Normal 1 and Chi-squared 1 whereby \\Sigma X^2
type: Transformation
distribution relationship from: Standard Normal 1
distribution relationship to: Chi-square
distribution relationship definition in Latex: \Sigma X^2
distribution relationship pair expression Latex: StandardNormal1(0,1) \rightarrow ChiSquared1(k)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardnormalChisquare.pdf}

mean of Negative Binomial 5
type: mathematical object
has name: mean of Negative Binomial 5
has Latex expression: \alpha / \beta

logit of success probability in each trial of Binomial-2
type: mathematical object
has name: logit of success probability in each trial of Binomial-2
has code name: Binomial2.logitProbability
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R
has short name: logit of success probability in each trial
has short code name: logitProbability

mean of logx of Log-Normal-1
type: mathematical object
has name: mean of log(x) of Log-Normal-1
has code name: LogNormal1.meanLog
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean of log(x)
has short code name: meanLog

relationship between Log-Normal 6 and Log-Normal 3 whereby m = m, \\sigma=\\log\\sigma_g
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 3
distribution relationship definition in Latex: m = m, \sigma=\log(\sigma_g)
distribution relationship pair expression Latex: LogNormal6(m,\sigma_g) \rightarrow LogNormal3(m,\sigma)
distribution relationship reference: ProbOnto spec

PDF of Multivariate Student T 2
type: probability density function
has name: PDF of Multivariate (Student) T 2
has Latex expression: \frac{\Gamma((k+d)/2)}{\Gamma(k/2) k^{d/2} \pi^{d/2}}|T|^{1/2} \Big[ 1 + \frac{1}{k}(x-\mu)' T (x-\mu) \Big]^{-(k+d)/2}

PROB k0000816
type: Transformation
label: relationship between Scaled Inverse Chi-Square and Scaled Inverse Chi-Square whereby \\text{If } X \\sim ScaledInverseChiSquare1\\nu,\\sigma \\text{ then } kX \\sim ScaledInverseChiSquare1\\nu, k\\sigma
distribution relationship from: Scaled Inverse Chi-Square
distribution relationship to: Scaled Inverse Chi-Square
distribution relationship definition in Latex: \text{If } X \sim ScaledInverseChiSquare1(\nu,\sigma) \text{ then } kX \sim ScaledInverseChiSquare1(\nu, k\sigma)
distribution relationship pair expression Latex: ScaledInverseChiSquare1(\nu,\sigma) \rightarrow ScaledInverseChiSquare1(\nu,\sigma)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution}

variance of Dagum 1
type: mathematical object
has name: variance of Dagum 1
has Latex expression: \begin{cases} -\frac{b^2}{a^2} \left(2 a \frac{\Gamma\left(-\tfrac{2}{a}\right) \, \Gamma\left(\tfrac{2}{a} + p\right)}{\Gamma\left(p\right)} + \left( \frac{\Gamma\left(-\tfrac{1}{a}\right) \Gamma\left(\tfrac{1}{a} + p\right)}{\Gamma\left(p\right)} \right)^2\right) & \text{if}\ a>2 \\ \text{Indeterminate} & \text{otherwise} \end{cases}

shape parameter of Kumaraswamy-1
type: mathematical object
has name: shape parameter of Kumaraswamy-1
has code name: Kumaraswamy1.shape2
has Latex symbol: b
parameter has definition expressed in plain text: b > 0
has short name: shape parameter
has short code name: shape2

relationship between Poisson 2 and Poisson 1 whereby \\lambda = \\exp\\alpha
type: Transformation
distribution relationship from: Poisson 2
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \lambda = \exp(\alpha)
distribution relationship pair expression Latex: Poisson2(\alpha) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{stan-manual:2015b}

index parameter of Negative-Binomial-6
type: mathematical object
has name: index parameter of Negative-Binomial-6
has code name: NegativeBinomial6.dispersion
has Latex symbol: \phi
parameter has definition expressed in plain text: \phi \in R, \phi > 0
has short name: index parameter
has short code name: dispersion

number of successes of Negative-Binomial-1
type: mathematical object
has name: number of successes of Negative-Binomial-1
has code name: NegativeBinomial1.numberOfSuccesses
has Latex symbol: r
parameter has definition expressed in plain text: r > 0, r \in N
has short name: number of successes
has short code name: numberOfSuccesses

location parameter of Johnson-SN-1
type: mathematical object
has name: location parameter of Johnson-SN-1
has code name: JohnsonSN1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location parameter
has short code name: location

variance of Von Mises 1
type: mathematical object
has name: variance of Von Mises 1
has Latex expression: 1 - \frac{I_1(\kappa)}{I_1(0)} \text{ (circular variance)}

median of Poisson 1
type: mathematical object
has name: median of Poisson 1
has Latex expression: \approx\lfloor\lambda+1/3-0.02/\lambda\rfloor

PMF of Binomial 1
type: probability mass function
has name: PMF of Binomial 1
has Latex expression: {n \choose k}\, p^k (1-p)^{n-k}
has R code expression: choose(n,k) * p^k*(1-p)^(n-k)

PDF of Johnson SB 1
type: probability density function
has name: PDF of Johnson SB 1
has Latex expression: \frac{e^{-\frac12 \left( \gamma + \delta \log\left[\frac{x-\mu}{-x+\mu+\sigma}\right]\right)^2} \delta \sigma}{\sqrt{2\pi} (x-\mu)(-x+\mu+\sigma)}
has R code expression: exp(-1/2*(gamma+delta*log((x-mu)/(-x+mu+sigma)))^2)*delta*sigma/(sqrt(2*pi)*(x-mu)*(-x+mu+sigma))

scale of Power-Normal-1
type: mathematical object
has name: scale of Power-Normal-1
has code name: PowerNormal1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma \in R, \sigma > 0
has short name: scale
has short code name: scale

Poisson intensity of Zero-inflated-Poisson-1
type: mathematical object
has name: Poisson intensity of Zero-inflated-Poisson-1
has code name: ZeroInflatedPoisson1.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in R, \lambda > 0
has short name: Poisson intensity
has short code name: rate

HF of Uniform Discrete 2
type: hazard function
has name: HF of Uniform Discrete 2
has Latex expression: \frac{1}{n-k}
has R code expression: 1/(n-k)

degrees of freedom of Noncentral-F-1
type: mathematical object
has name: degrees of freedom of Noncentral-F-1
has code name: NoncentralF1.degreesOfFreedom1
has Latex symbol: m
parameter has definition expressed in plain text: m \in N
has short name: degrees of freedom
has short code name: degreesOfFreedom1

variance of Inverse Chi-Square
type: mathematical object
has name: variance of Inverse Chi-Square
has Latex expression: \frac{2}{(\nu - 2)^2(\nu-4)} \text{ for } \nu>4

relationship between Negative Binomial 5 and Negative Binomial 1 whereby r = \\alpha, p = \\beta / 1 + \\beta
type: Reparameterisation
distribution relationship from: Negative Binomial 5
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: r = \alpha, p = \beta / (1 + \beta)
distribution relationship pair expression Latex: NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial1(r,p)
distribution relationship reference: ProbOnto spec

CDF of Negative Binomial 5
type: cumulative density function
has name: CDF of Negative Binomial 5
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

variance of Log-Normal 1
type: mathematical object
has name: variance of Log-Normal 1
has Latex expression: (e^{\sigma^2}\!\!-1) e^{2\mu+\sigma^2}

CDF of Beta Negative Binomial1
type: cumulative density function
has name: CDF of Beta Negative Binomial1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

mean of Negative Binomial 3
type: mathematical object
has name: mean of Negative Binomial 3
has Latex expression: \mu

Johnson SB 1
type: probability distribution
has name: Johnson SB 1
has code name: JohnsonSB1
distribution has PDF: PDF of Johnson SB 1
distribution has CDF: CDF of Johnson SB 1
has variate symbol in Latex: x
has Latex expression of support: x \in (\mu,\mu+\sigma
distribution has parameter: location parameter of Johnson-SB-1
distribution has parameter: shape parameter of Johnson-SB-1
distribution has parameter: scale parameter of Johnson-SB-1

relationship between Binomial 1 and Poisson 1 whereby \\lambda = np, n \\rightarrow \\infty
type: Limiting
distribution relationship from: Binomial 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \lambda = np, n \rightarrow \infty
distribution relationship pair expression Latex: Binomial1(n,p) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialPoisson.pdf}

log mean of Negative-Binomial-6
type: mathematical object
has name: log mean of Negative-Binomial-6
has code name: NegativeBinomial6.logMean
has Latex symbol: \eta
parameter has definition expressed in plain text: \eta \in R
has short name: log mean
has short code name: logMean

Multivariate Student T 2
type: probability distribution
has name: Multivariate (Student) T 2
has code name: MultivariateStudentT2
distribution has PDF: PDF of Multivariate Student T 2
has variate symbol in Latex: x
has Latex expression of support: x \in R^d, k\geq 2
distribution has parameter: precision matrix of Multivariate-Student-T-2
distribution has parameter: degrees of freedom of Multivariate-Student-T-2
distribution has parameter: location of Multivariate-Student-T-2

degrees of freedom of Multivariate-Student-T-1
type: mathematical object
has name: degrees of freedom of Multivariate-Student-T-1
has code name: MultivariateStudentT1.degreesOfFreedom
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu \ge 2
has short name: degrees of freedom
has short code name: degreesOfFreedom

Inverse Binomial 1
type: probability distribution
has name: Inverse Binomial 1
has code name: InverseBinomial1
distribution has PMF: PMF of Inverse Binomial 1
distribution has mean: mean of Inverse Binomial 1
distribution has variance: variance of Inverse Binomial 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{0,1,2,3,\dots\}
distribution has parameter: probability of Inverse-Binomial-1
distribution has parameter: index parameter of Inverse-Binomial-1

mode of Dagum 1
type: mathematical object
has name: mode of Dagum 1
has Latex expression: b{\left( \frac{ap-1}{a+1} \right)}^{\tfrac{1}{a}}

relationship between Zero-Inflated Generalized Poisson 1 and Generalized Poisson 3 whereby p0=0
type: Special case
distribution relationship from: Zero-Inflated Generalized Poisson 1
distribution relationship to: Generalized Poisson 3
distribution relationship definition in Latex: p0=0
distribution relationship pair expression Latex: ZeroInflatedGeneralizedPoisson1(\mu,\alpha,p0) \rightarrow GeneralizedPoisson3(\mu,\alpha)
distribution relationship reference: \cite{famoye2006zero}

Poisson intensity of Poisson-1
type: mathematical object
has name: Poisson intensity of Poisson-1
has code name: Poisson1.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in R, \lambda > 0
has short name: Poisson intensity
has short code name: rate

CDF of Birnbaum-Saunders 1
type: cumulative density function
has name: CDF of Birnbaum-Saunders 1
has Latex expression: \int_0^x f(x), \text { with f the PDF}
has R code expression: cumsum(PDF*rep(stepSize,length(PDF)))

variance of Zero-inflated Poisson 1
type: mathematical object
has name: variance of Zero-inflated Poisson 1
has Latex expression: \lambda (1-\pi) (1 + \lambda \pi)

PROB k0000967
type: Limiting
label: relationship between Hypergeometric 1 and Poisson 1 whereby X \\sim Hypergeometric1N, K, n \\Rightarrow Y\\sim Poisson1\\lambda \\text{ as K, N and n tend to infinity for } \\\\ K/N \\text{ small and } nK/N \\rightarrow \\lambda
distribution relationship from: Hypergeometric 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: X \sim Hypergeometric1(N, K, n) \Rightarrow Y\sim Poisson1(\lambda) \text{ as K, N and n tend to infinity for } \\ K/N \text{ small and } nK/N \rightarrow \lambda
distribution relationship pair expression Latex: Hypergeometric1(N,K,n) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{forbes2011statistical}

mode of Von Mises 1
type: mathematical object
has name: mode of Von Mises 1
has Latex expression: \mu

inverse scale matrix of Wishart-2
type: mathematical object
has name: inverse scale matrix of Wishart-2
has code name: Wishart2.inverseScaleMatrix
has Latex symbol: R
parameter has definition expressed in plain text: p\times p - \text{symmetric, positive definite matrix}
has short name: inverse scale matrix
has short code name: inverseScaleMatrix

scale parameter of Johnson-SN-1
type: mathematical object
has name: scale parameter of Johnson-SN-1
has code name: JohnsonSN1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: scale parameter
has short code name: scale

PROB k0000814
type: Limiting
label: relationship between Student's t-distribution 1 and Standard Normal 1 whereby X\\sim StudentT1\\nu \\Rightarrow StandardNormal10,1 \\text{ as } \\nu \\text{ tends to infinity.}\\\\ \\text{ The approximation is reasonable for } \\nu\\ge 30
distribution relationship from: t-distribution
distribution relationship to: Standard Normal 1
distribution relationship definition in Latex: X\sim StudentT1(\nu) \Rightarrow StandardNormal1(0,1) \text{ as } \nu \text{ tends to infinity.}\\ \text{ The approximation is reasonable for } \nu\ge 30
distribution relationship pair expression Latex: StudentT1(\nu) \rightarrow StandardNormal1(0,1)
distribution relationship reference: \cite{forbes2011statistical}

noncentrality parameter of Noncentral-F-1
type: mathematical object
has name: noncentrality parameter of Noncentral-F-1
has code name: NoncentralF1.noncentrality
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \geq 0
has short name: noncentrality parameter
has short code name: noncentrality

variance of Uniform Discrete 2
type: mathematical object
has name: variance of Uniform Discrete 2
has Latex expression: \frac{n(n+2)}{12}

CDF of Power-Normal 1
type: cumulative density function
has name: CDF of Power-Normal 1
has Latex expression: \begin{cases} \frac{1}{K} (\Phi(Z) - \Phi(-T) ) & \text{for } \lambda > 0 \\ \Phi(Z) & \text{for } \lambda = 0 \\ \frac{1}{K} \Phi & \text{for } \lambda < 0 \end{cases}\\ \text{with } Z = \frac{(x^\lambda - 1) / \lambda-\mu}{\sigma} \quad \text{and} \quad T = 1/(\lambda \sigma) + \mu/\sigma
has R code expression: CDF = function(x,lambda,mu,sigma) { if (lambda > 0) { CDF = 1/K(lambda,mu,sigma) * ( PHI(Zfunc(x,lambda,mu,sigma)) - PHI(-Tfunc(lambda,mu,sigma)) ) } else if (lambda==0) { CDF = PHI(Zfunc(x,lambda,mu,sigma)) } else if (lambda < 0) { CDF = 1/K(lambda,mu,sigma) * PHI(Zfunc(x,lambda,mu,sigma)) } } Zfunc <- function(x,lambda,mu,sigma) { Zfunc = (BCtrafo(x,lambda)-mu) / sigma }

CDF of Inverse Chi-Square
type: cumulative density function
has name: CDF of Inverse Chi-Square
has Latex expression: \Gamma\Big(\frac{\nu}{2},\frac{1}{2x}\Big)\Big/\Gamma\Big(\frac{\nu}{2}\Big)
has R code expression: Igamma(nu/2,1/2/x, lower=F) / gamma(nu/2)

Negative Binomial 3
type: probability distribution
has name: Negative Binomial 3
has code name: NegativeBinomial3
distribution has PMF: PMF of Negative Binomial 3
distribution has mean: mean of Negative Binomial 3
distribution has variance: variance of Negative Binomial 3
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: mean of Negative-Binomial-3
distribution has parameter: index parameter of Negative-Binomial-3

PMF of Negative Binomial 5
type: probability mass function
has name: PMF of Negative Binomial 5
has Latex expression: \binom {k+\alpha-1}{\alpha-1} \Big(\frac{\beta}{\beta + 1} \Big)^{\alpha} \Big(\frac{1}{\beta + 1} \Big)^{k}
has R code expression: choose(k+alpha-1,alpha-1) * (beta / (beta + 1))^alpha * (1 / (beta + 1))^k

relationship between Weibull 1 and Rayleigh 1 whereby k=2, \\lambda=\\sqrt{2} \\sigma
type: Reparameterisation
distribution relationship from: Weibull 1
distribution relationship to: Rayleigh 1
distribution relationship definition in Latex: k=2, \lambda=\sqrt{2} \sigma
distribution relationship pair expression Latex: Weibull1(\lambda,k) \rightarrow Rayleigh1(\sigma)
distribution relationship reference: ProbOnto spec

variance of Negative Binomial 6
type: mathematical object
has name: variance of Negative Binomial 6
has Latex expression: \exp(\eta) + \exp(\eta)^2 / \phi

covariance matrix of Multivariate-Student-T-1
type: mathematical object
has name: covariance matrix of Multivariate-Student-T-1
has code name: MultivariateStudentT1.covarianceMatrix
has Latex symbol: \Sigma
parameter has definition expressed in plain text: \Sigma, \text{ positive-definite real } p\times p \text{ matrix}
has short name: covariance matrix
has short code name: covarianceMatrix

scale of Gompertz-1
type: mathematical object
has name: scale of Gompertz-1
has code name: Gompertz1.scale
has Latex symbol: b
parameter has definition expressed in plain text: b > 0
has short name: scale
has short code name: scale

median of Dagum 1
type: mathematical object
has name: median of Dagum 1
has Latex expression: b{\left(-1+2^{\tfrac{1}{p}}\right)}^{-\tfrac{1}{a}}

Poisson 2
type: probability distribution
has name: Poisson 2
has code name: Poisson2
distribution has PMF: PMF of Poisson 2
distribution has CDF: CDF of Poisson 2
distribution has mean: mean of Poisson 2
distribution has variance: variance of Poisson 2
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: log Poisson intensity of Poisson-2

number of elements of Zipf-1
type: mathematical object
has name: number of elements of Zipf-1
has code name: Zipf1.numberOfElements
has Latex symbol: n
parameter has definition expressed in plain text: n \in N^+
has short name: number of elements
has short code name: numberOfElements

CDF of Binomial 2
type: cumulative density function
has name: CDF of Binomial 2
has Latex expression: I_{1-logit^{-1}(\alpha)}(n - k, 1 + k)
has R code expression: invLogit = function(x) { exp(x)/(1+exp(x)) } \\ Rbeta(1-invLogit(alpha), n-k, 1+k)

PMF of Beta Negative Binomial1
type: probability mass function
has name: PMF of Beta Negative Binomial1
has Latex expression: \frac{{n-1+x \choose x}{B(n+\alpha,\beta+x)}}{B(\alpha,\beta)}
has R code expression: choose(n-1+x,x)*beta(n+alpha,beta+x)/beta(alpha,beta)

median of Von Mises 1
type: mathematical object
has name: median of Von Mises 1
has Latex expression: \mu

mean of Zero-inflated Poisson 1
type: mathematical object
has name: mean of Zero-inflated Poisson 1
has Latex expression: (1-\pi) \lambda

variance of Poisson 1
type: mathematical object
has name: variance of Poisson 1
has Latex expression: \lambda

variance of Noncentral F 1
type: mathematical object
has name: variance of Noncentral F 1
has Latex expression: \left(\frac{n}{m}\right)^2 \frac{2}{(n-2)(n-4)}\left[\frac{(\lambda+m)^2}{n-2} +2\lambda + m \right]

CDF of Dagum 1
type: cumulative density function
has name: CDF of Dagum 1
has Latex expression: \left( 1+{\left(\frac{x}{b}\right)}^{-a} \right)^{-p}
has R code expression: (1+(x/b)^(-a))^(-p)

relationship between Student's t-distribution 1 and F 1 whereby \\text{If } X\\sim StudentT1\\nu \\Rightarrow Y=X^2 \\sim F1,\\nu
type: Transformation
distribution relationship from: t-distribution
distribution relationship to: F 1
distribution relationship definition in Latex: \text{If } X\sim StudentT1(\nu) \Rightarrow Y=X^2 \sim F(1,\nu)
distribution relationship pair expression Latex: StudentT1(\nu) \rightarrow F1(n_1,n_2)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TF.pdf}

degrees of freedom of Wishart-2
type: mathematical object
has name: degrees of freedom of Wishart-2
has code name: Wishart2.degreesOfFreedom
has Latex symbol: k
has short name: degrees of freedom
has short code name: degreesOfFreedom

Muth 1
type: probability distribution
has name: Muth 1
has code name: Muth1
distribution has PDF: PDF of Muth 1
distribution has CDF: CDF of Muth 1
distribution has SF: SF of Muth 1
distribution has HF: HF of Muth 1
distribution has mean: mean of Muth 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape parameter of Muth-1

mean of Uniform Discrete 2
type: mathematical object
has name: mean of Uniform Discrete 2
has Latex expression: \frac{n}{2}

transforming parameter of Power-Normal-1
type: mathematical object
has name: transforming parameter of Power-Normal-1
has code name: PowerNormal1.shape
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in R
has short name: transforming parameter
has short code name: shape

relationship between Zero-Inflated Negative Binomial 1 and Negative Binomial 2 whereby p0=0
type: Special case
distribution relationship from: Zero-Inflated Negative Binomial 1
distribution relationship to: Negative Binomial 2
distribution relationship definition in Latex: p0=0
distribution relationship pair expression Latex: ZeroInflatedNegativeBinomial1(\lambda,\tau,p0) \rightarrow NegativeBinomial2(\lambda,\tau)
distribution relationship reference: \cite{Troconiz:2009fv}

Negative Binomial 5
type: probability distribution
has name: Negative Binomial 5
has code name: NegativeBinomial5
distribution has PMF: PMF of Negative Binomial 5
distribution has CDF: CDF of Negative Binomial 5
distribution has mean: mean of Negative Binomial 5
distribution has variance: variance of Negative Binomial 5
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: shape of Negative-Binomial-5
distribution has parameter: inverse scale of Negative-Binomial-5

mean of Inverse Chi-Square
type: mathematical object
has name: mean of Inverse Chi-Square
has Latex expression: \frac{1}{\nu - 2} \text{ for } \nu>2

shape of Gompertz-1
type: mathematical object
has name: shape of Gompertz-1
has code name: Gompertz1.shape
has Latex symbol: \eta
parameter has definition expressed in plain text: \eta > 0
has short name: shape
has short code name: shape

mean of Dagum 1
type: mathematical object
has name: mean of Dagum 1
has Latex expression: \begin{cases} -\frac{b}{a}\frac{\Gamma\left(-\tfrac{1}{a}\right)\Gamma\left(\tfrac{1}{a}+p\right)}{\Gamma(p)} & \text{if}\ a>1 \\ \text{Indeterminate} & \text{otherwise} \end{cases}

relationship between Log-Normal 1 and Log-Normal 5 whereby \\mu = \\mu, \\tau = 1 / \\sigma^2
type: Reparameterisation
distribution relationship from: Log-Normal 1
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu = \mu, \tau = 1 / \sigma^2
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow LogNormal5(\mu,\tau)
distribution relationship reference: ProbOnto spec

location of Multivariate-Student-T-1
type: mathematical object
has name: location of Multivariate-Student-T-1
has code name: MultivariateStudentT1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu = [\mu_1, \dots, \mu_p]^T, \mu_i \in R
has short name: location
has short code name: mean

mean of Negative Binomial 6
type: mathematical object
has name: mean of Negative Binomial 6
has Latex expression: \exp(\eta)

relationship between Conway-Maxwell-Poisson 1 and Poisson 1 whereby \\text{For } \\nu = 1 \\text{ the sum has a Poisson distribution with parameter } n\\lambda
type: Transformation
distribution relationship from: Conway-Maxwell-Poisson 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \text{For } \nu = 1 \text{ the sum has a Poisson distribution with parameter } n\lambda
distribution relationship pair expression Latex: ConwayMaxwellPoisson1(\lambda,\nu) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{shmueli2005useful}

PMF of Negative Binomial 3
type: probability mass function
has name: PMF of Negative Binomial 3
has Latex expression: \binom {k+\phi-1}k \Big(\frac{\phi}{\mu + \phi} \Big)^{\phi} \Big(\frac{\mu}{\mu + \phi} \Big)^{k}
has R code expression: choose(k+phi-1,k) * (phi / (mu + phi))^phi * (mu / (mu + phi))^k

shape of Zipf-1
type: mathematical object
has name: shape of Zipf-1
has code name: Zipf1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R, \alpha > 0
has short name: shape
has short code name: shape

shape of Log-Normal-1
type: mathematical object
has name: shape of Log-Normal-1
has code name: LogNormal1.stdevLog
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: shape
has short code name: stdevLog

Beta Negative Binomial1
type: probability distribution
label: Beta Pascal
has name: Beta Negative Binomial1
has code name: BetaNegativeBinomial1
distribution has PMF: PMF of Beta Negative Binomial1
distribution has CDF: CDF of Beta Negative Binomial1
distribution has mean: mean of Beta Negative Binomial1
has variate symbol in Latex: x
has Latex expression of support: x \in \{0,\dots,n\}
distribution has parameter: number of successes of Beta-Negative-Binomial1
distribution has parameter: shape parameter of Beta-Negative-Binomial1

number of trials of Binomial-2
type: mathematical object
has name: number of trials of Binomial-2
has code name: Binomial2.numberOfTrials
has Latex symbol: n
parameter has definition expressed in plain text: n \in N, n \ge 0
has short name: number of trials
has short code name: numberOfTrials

median of Log-Normal 4
type: mathematical object
has name: median of Log-Normal 4
has Latex expression: m

Sinh-Arcsinh 2
type: probability distribution
label: SHASH
has name: Sinh-Arcsinh 2
has code name: SinhArcsinh2
distribution has PDF: PDF of Sinh-Arcsinh 2
distribution has CDF: CDF of Sinh-Arcsinh 2
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: kurtosis of Sinh-Arcsinh-2
distribution has parameter: location of Sinh-Arcsinh-2
distribution has parameter: skewness of Sinh-Arcsinh-2
distribution has parameter: scale of Sinh-Arcsinh-2

mean of Normal-1
type: mathematical object
has name: mean of Normal-1
has code name: Normal1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: mean

degrees of freedom of NoncentralT
type: mathematical object
has name: degrees of freedom of NoncentralT
has code name: NoncentralT1.degreesOfFreedom
has Latex symbol: n
parameter has definition expressed in plain text: n \in N
has short name: degrees of freedom
has short code name: degreesOfFreedom

relationship between Generalized Gamma 2 and Weibull 1 whereby c=1, a=0, b=\\lambda
type: Reparameterisation
distribution relationship from: Generalized Gamma 2
distribution relationship to: Weibull 1
distribution relationship definition in Latex: c=1, a=0, b=\lambda
distribution relationship pair expression Latex: GeneralizedGamma2(a,b,c,k) \rightarrow Weibull1(\lambda,k)
distribution relationship reference: \cite{forbes2011statistical}

variance of Multivariate Student T 1
type: mathematical object
has name: variance of Multivariate (Student) T 1
has Latex expression: \begin{cases} \frac{\nu}{\nu-2} \Sigma & \text{for }\nu > 2 \\ undefined & \text{else} \end{cases}

PMF of Double Poisson 1
type: probability mass function
has name: PMF of Double Poisson 1
has Latex expression: K(\mu,\phi) \phi^{1/2} \exp(-\phi \mu) \frac{\exp(-y) y^y}{y!}\Big(\frac{e \mu}{y}\Big)^{\phi y} \\ \text{ with } \frac{1}{K(\mu,\phi)} \approx 1 + \frac{1-\phi}{12\phi \mu} \Big( 1 + \frac{1}{\phi \mu} \Big)
has R code expression: K(mu,phi) * phi^(1/2) * exp(-phi*mu) * (exp(-y)*y^y)/factorial(y) * (exp(1)*mu/y)^(phi*y) \\ # with K(mu,phi) = 1 / (1 + (1-phi)/(12*mu*phi)*(1 + 1/(mu*phi)) )

minimum of Log-Uniform-1
type: mathematical object
has name: minimum of Log-Uniform-1
has code name: LogUniform1.minimum
has Latex symbol: min
parameter has definition expressed in plain text: min>0
has short name: minimum
has short code name: minimum

Inverted-chi-square
type: probability distribution
label: Inverse Chi-Square
has name: Inverse Chi-Square
has code name: InverseChiSquare1
distribution has PDF: PDF of Inverse Chi-Square
distribution has CDF: CDF of Inverse Chi-Square
distribution has mean: mean of Inverse Chi-Square
distribution has mode: mode of Inverse Chi-Square
distribution has variance: variance of Inverse Chi-Square
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: degrees of freedom of Inverse-Chi-Square

mean of Log-Normal 1
type: mathematical object
has name: mean of Log-Normal 1
has Latex expression: e^{\mu+\sigma^2/2}

location parameter of Asymmetric-Laplace-1
type: mathematical object
has name: location parameter of Asymmetric-Laplace-1
has code name: AsymmetricLaplace1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location parameter
has short code name: location

relationship between Binomial 2 and Binomial 1 whereby p = \\exp\\alpha / 1 + \\exp\\alpha
type: Reparameterisation
distribution relationship from: Binomial distribution with logit parameterisation
distribution relationship to: Binomial 1
distribution relationship definition in Latex: p = \exp(\alpha) / (1 + \exp(\alpha))
distribution relationship pair expression Latex: Binomial2(n,\alpha) \rightarrow Binomial1(n,p)
distribution relationship reference: ProbOnto spec

variance of Erlang 1
type: mathematical object
has name: variance of Erlang 1
has Latex expression: b^2 c

HF of Burr 1
type: hazard function
has name: HF of Burr 1
has Latex expression: \frac{\frac{kc}{\alpha}(x/\alpha)^{c-1}}{1+(x/\alpha)^c}
has R code expression: (k*c)/alpha * (x/alpha)^(c-1) / (1+(x/alpha)^c)

relationship between Log-Normal 5 and Log-Normal 2 whereby \\mu = \\mu, v = 1 / \\tau
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 2
distribution relationship definition in Latex: \mu = \mu, v = 1 / \tau
distribution relationship pair expression Latex: LogNormal5(\mu,\tau) \rightarrow LogNormal2(\mu,v)
distribution relationship reference: ProbOnto spec

median / geometric mean of Log-Normal-3
type: mathematical object
has name: median / geometric mean of Log-Normal-3
has code name: LogNormal3.median
has Latex symbol: m
parameter has definition expressed in plain text: m>0
has short name: median / geometric mean
has short code name: median

PROB k0001450
type: Special case
label: relationship between Amoroso 1 and Chi-squared 1 whereby \\text{ChiSquared1}k \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a = 0, \\theta=2, \\alpha=k/2, \\beta=1
distribution relationship from: Amoroso 1
distribution relationship to: Chi-square
distribution relationship definition in Latex: \text{ChiSquared1}(k) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a = 0, \theta=2, \alpha=k/2, \beta=1
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow ChiSquared1(k)
distribution relationship reference: \cite{crooks2010amoroso}

SF of Log-Logistic 2
type: survival function
has name: SF of Log-Logistic 2
has Latex expression: \frac{1}{1+(\lambda x)^\kappa}
has R code expression: 1/(1+(lambda*x)^kappa)

degrees of freedom of Chi-squared-1
type: mathematical object
has name: degrees of freedom of Chi-squared-1
has code name: ChiSquared1.degreesOfFreedom
has Latex symbol: k
parameter has definition expressed in plain text: k \in N
has short name: degrees of freedom
has short code name: degreesOfFreedom

relationship between Negative Binomial 5 and Negative Binomial 2 whereby \\lambda = \\alpha / \\beta, \\tau = 1 / \\alpha
type: Reparameterisation
distribution relationship from: Negative Binomial 5
distribution relationship to: Negative Binomial 2
distribution relationship definition in Latex: \lambda = \alpha / \beta, \tau = 1 / \alpha
distribution relationship pair expression Latex: NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial2(\lambda, \tau)
distribution relationship reference: ProbOnto spec

scale parameter of Dagum-1
type: mathematical object
has name: scale parameter of Dagum-1
has code name: Dagum1.scale
has Latex symbol: b
parameter has definition expressed in plain text: b > 0
has short name: scale parameter
has short code name: scale

standard deviation of Normal-1
type: mathematical object
has name: standard deviation of Normal-1
has code name: Normal1.stdev
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: standard deviation
has short code name: stdev

PMF of Binomial 2
type: probability mass function
has name: PMF of Binomial 2
has Latex expression: {n \choose k}\, \big(logit^{-1}(\alpha)\big)^k \big(1-logit^{-1}(\alpha)\big)^{n-k}
has R code expression: invLogit = function(x) { exp(x)/(1+exp(x)) } choose(n,k)*invLogit(alpha)^k*(1-invLogit(alpha))^(n-k)

mean of Log-Normal 4
type: mathematical object
has name: mean of Log-Normal 4
has Latex expression: m \sqrt{cv^2 + 1}

noncentrality parameter of NoncentralT
type: mathematical object
has name: noncentrality parameter of NoncentralT
has code name: NoncentralT1.noncentrality
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \geq 0
has short name: noncentrality parameter
has short code name: noncentrality

maximum of Log-Uniform-1
type: mathematical object
has name: maximum of Log-Uniform-1
has code name: LogUniform1.maximum
has Latex symbol: max
parameter has definition expressed in plain text: max \geq min
has short name: maximum
has short code name: maximum

mode of Multivariate Student T 1
type: mathematical object
has name: mode of Multivariate (Student) T 1
has Latex expression: \mu

CDF of Double Poisson 1
type: cumulative density function
has name: CDF of Double Poisson 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

PDF of Inverse Chi-Square
type: probability density function
has name: PDF of Inverse Chi-Square
has Latex expression: \frac{2^{-\nu/2}}{\Gamma(\nu/2)} x^{-(\nu/2+1)} \exp\Big(-\frac{1}{2x}\Big)
has R code expression: 2^(-nu/2)/gamma(nu/2)*x^(-(nu/2+1))*exp(-1/(2*x))

CDF of Log-Normal 1
type: cumulative density function
has name: CDF of Log-Normal 1
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\mu}{\sqrt{2}\sigma}\Big]
has R code expression: 1/2 + 1/2 *erf( (log(x)-mu)/(sqrt(2)*sigma) )

CDF of Burr 1
type: cumulative density function
has name: CDF of Burr 1
has Latex expression: 1 - \frac{1}{\Big[1+(x/\alpha)^c\Big]^k}
has R code expression: 1 - 1/(1+(x/alpha)^c)^k

relationship between Generalized Poisson 1 and Poisson 1 whereby \\delta = 0, \\theta=\\mu
type: Special case
distribution relationship from: Generalized Poisson 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: \delta = 0, \theta=\mu
distribution relationship pair expression Latex: GeneralizedPoisson1(\theta,\delta) \rightarrow Poisson1(\lambda)
distribution relationship reference: \cite{Yang:2007vn}

variance of Chi-squared 1
type: mathematical object
has name: variance of Chi-squared 1
has Latex expression: 2k

scale parameter of Asymmetric-Laplace-1
type: mathematical object
has name: scale parameter of Asymmetric-Laplace-1
has code name: AsymmetricLaplace1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma \geq 0
has short name: scale parameter
has short code name: scale

SF of Gamma 1
type: survival function
has name: SF of Gamma 1
has Latex expression: 1 - \frac{\Gamma(\beta,x/\alpha)}{\Gamma(\beta)}

Level start of Trapezoidal-1
type: mathematical object
has name: Level start of Trapezoidal-1
has code name: Trapezoidal1.levelStart
has Latex symbol: b
parameter has definition expressed in plain text: a \leq b < c
has short name: Level start
has short code name: levelStart

relationship between Power-Normal 1 and Truncated Normal 1 whereby PowerNormal1 \\text{ converges to } TruncatedNormal1 \\text{ when } \\lambda \\rightarrow 1
type: Special case
distribution relationship from: Power-Normal 1
distribution relationship to: Truncated Normal 1
distribution relationship definition in Latex: PowerNormal1 \text{ converges to } TruncatedNormal1 \text{ when } \lambda \rightarrow 1
distribution relationship pair expression Latex: PowerNormal1(\lambda,\mu,\sigma) \rightarrow TruncatedNormal1(\mu,\sigma,a,b)
distribution relationship reference: \cite{LavielleBook:2014}

variance of Log-Normal 3
type: mathematical object
has name: variance of Log-Normal 3
has Latex expression: m^2 e^{\sigma^2} [e^{\sigma^2}-1]

mean of NoncentralT
type: mathematical object
has name: mean of NoncentralT
has Latex expression: sqrt(n/2)*gamma((n-1)/2) / gamma(n/2)*delta

parameter has mathematical type
type: DatatypeProperty

relationship between Log-Normal 3 and Log-Normal 5 whereby \\mu = \\logm, \\tau = 1 / \\sigma^2
type: Reparameterisation
distribution relationship from: Log-Normal 3
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu = \log(m), \tau = 1 / \sigma^2
distribution relationship pair expression Latex: LogNormal3(m,\sigma) \rightarrow LogNormal5(\mu,\tau)
distribution relationship reference: ProbOnto spec

mode of Log-Normal 1
type: mathematical object
has name: mode of Log-Normal 1
has Latex expression: e^{\mu-\sigma^2}

shape parameter of Dagum-1
type: mathematical object
has name: shape parameter of Dagum-1
has code name: Dagum1.shape2
has Latex symbol: a
parameter has definition expressed in plain text: a > 0
has short name: shape parameter
has short code name: shape2

variance of Log-Normal 4
type: mathematical object
has name: variance of Log-Normal 4
has Latex expression: m^2 (cv^2+1) cv^2

Upper bound of Trapezoidal-1
type: mathematical object
has name: Upper bound of Trapezoidal-1
has code name: Trapezoidal1.upperBound
has Latex symbol: d
parameter has definition expressed in plain text: c \leq d
has short name: Upper bound
has short code name: upperBound

mode of Normal 1
type: mathematical object
has name: mode of Normal 1
has Latex expression: \mu

mean of Log-Uniform 1
type: mathematical object
has name: mean of Log-Uniform 1
has Latex expression: \frac{max-min}{\log(max) - \log(min)}

median of Multivariate Student T 1
type: mathematical object
has name: median of Multivariate (Student) T 1
has Latex expression: \mu

mean of Erlang 1
type: mathematical object
has name: mean of Erlang 1
has Latex expression: bc

PDF of Burr 1
type: probability density function
has name: PDF of Burr 1
has Latex expression: \frac{\frac{kc}{\alpha}(x/\alpha)^{c-1}}{\Big[1+(x/\alpha)^c\Big]^{k+1}}
has R code expression: k*c/alpha * (x/alpha)^(c-1) / (1+(x/alpha)^c)^(k+1)

mean of Asymmetric Laplace 1
type: mathematical object
has name: mean of Asymmetric Laplace 1
has Latex expression: \mu

location of Laplace-2
type: mathematical object
has name: location of Laplace-2
has code name: Laplace2.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

shape of Weibull-1
type: mathematical object
has name: shape of Weibull-1
has code name: Weibull1.shape
has Latex symbol: k
parameter has definition expressed in plain text: k\in (0, +\infty)
has short name: shape
has short code name: shape

CDF of Binomial 1
type: cumulative density function
has name: CDF of Binomial 1
has Latex expression: I_{1-p}(n - k, 1 + k)
has R code expression: Rbeta(1-p, n-k, 1+k)

Level end of Trapezoidal-1
type: mathematical object
has name: Level end of Trapezoidal-1
has code name: Trapezoidal1.levelEnd
has Latex symbol: c
parameter has definition expressed in plain text: b < c \leq d
has short name: Level end
has short code name: levelEnd

PDF of Logistic 1
type: probability density function
has name: PDF of Logistic 1
has Latex expression: \frac{e^{-\frac{x-\mu}{s}}} {s\left(1+e^{-\frac{x-\mu}{s}}\right)^2}
has R code expression: exp(-(x-mu)/s) / (s*(1+exp(-(x-mu)/s))^2)

Limiting
type: Class
subclass of: relationship between distributions

mode of Log-Normal 3
type: mathematical object
has name: mode of Log-Normal 3
has Latex expression: m/e^{\sigma^2}

PROB k0001452
type: Special case
label: relationship between Amoroso 1 and Chi 1 whereby \\text{Chi1}k \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a = 0, \\theta=\\sqrt{2}, \\alpha=k/2, \\beta=2
distribution relationship from: Amoroso 1
distribution relationship to: Chi 1
distribution relationship definition in Latex: \text{Chi1}(k) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a = 0, \theta=\sqrt{2}, \alpha=k/2, \beta=2
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Chi1(k)
distribution relationship reference: \cite{crooks2010amoroso}

standard deviation of Standard-Normal-1
type: mathematical object
has name: standard deviation of Standard-Normal-1
has code name: StandardNormal1.stdev
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma=1
has short name: standard deviation
has short code name: stdev

PDF of NoncentralT
type: probability density function
has name: PDF of NoncentralT
has Latex expression: \frac{e^{-\frac{\delta^2}{2}}}{\sqrt{n\pi} \;\Gamma\left(\frac{n}2\right)} \sum^{\infty}_{r=0} \frac{(t' \delta)^r}{r! n^{\frac{r}2}} \left(1+\frac{t'^2}{n}\right)^{-\frac{n+r+1}{2}}2^{\frac{r}{2}} \, \Gamma\left(\frac{n+r+1}{2}\right)
has R code expression: exp(-delta^2/2)/(sqrt(n*pi) * gamma(n/2)) * PDFsum(x,delta,n,lowerLimit,upperLimit) PDFsum = function(x,delta,n,fromValue,toValue) { PDF=array(0,length(x)); for(j in 1:length(x)) { for(r in fromValue:toValue) { PDF[j] = PDF[j] + (x[j]*delta)^r / (factorial(r)*n^(r/2)) * (1+x[j]^2/n)^(-(n+r+1)/2) * 2^(r/2) *gamma((n+r+1)/2) } } return(PDF) }

relationship between Weibull 1 and Weibull 2 whereby v=k, \\lambda = 1/\\lambda^k
type: Reparameterisation
distribution relationship from: Weibull 1
distribution relationship to: Weibull 2
distribution relationship definition in Latex: v=k, \lambda = 1/\lambda^k
distribution relationship pair expression Latex: Weibull1(\lambda,k) \rightarrow Weibull2(\lambda,v)
distribution relationship reference: ProbOnto spec

mode of Log-Normal 4
type: mathematical object
has name: mode of Log-Normal 4
has Latex expression: m/(cv^2+1)

shape parameter of Dagum-1
type: mathematical object
has name: shape parameter of Dagum-1
has code name: Dagum1.shape1
has Latex symbol: p
parameter has definition expressed in plain text: p > 0
has short name: shape parameter
has short code name: shape1

median of Log-Normal 1
type: mathematical object
has name: median of Log-Normal 1
has Latex expression: e^{\mu}

variance of Normal 1
type: mathematical object
has name: variance of Normal 1
has Latex expression: \sigma^2

mean of Multivariate Student T 1
type: mathematical object
has name: mean of Multivariate (Student) T 1
has Latex expression: \begin{cases} \mu & \text{for }\nu > 1 \\ undefined & \text{else} \end{cases}

Double Poisson 1
type: probability distribution
has name: Double Poisson 1
has code name: DoublePoisson1
distribution has PMF: PMF of Double Poisson 1
distribution has CDF: CDF of Double Poisson 1
distribution has mean: mean of Double Poisson 1
distribution has variance: variance of Double Poisson 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{1,2,3,\dots,n\}
distribution has parameter: dispersion of Double-Poisson-1
distribution has parameter: Poisson intensity of Double-Poisson-1

variance of Log-Uniform 1
type: mathematical object
has name: variance of Log-Uniform 1
has Latex expression: \frac{max^2-min^2}{2[\log(max) - \log(min)]} - \Big(\frac{max-min}{\log(max) - \log(min)}\Big)^2

mode of Erlang 1
type: mathematical object
has name: mode of Erlang 1
has Latex expression: b(c-1), c \ge 1

Singh-Maddala
type: probability distribution
label: Burr 1
label: Burr Type XII
has name: Burr 1
has code name: Burr1
distribution has PDF: PDF of Burr 1
distribution has CDF: CDF of Burr 1
distribution has SF: SF of Burr 1
distribution has HF: HF of Burr 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape parameter of Burr-1
distribution has parameter: scale parameter of Burr-1

variance of Asymmetric Laplace 1
type: mathematical object
has name: variance of Asymmetric Laplace 1
has Latex expression: \mu^2 + 2\sigma^2

Logistic 1
type: probability distribution
has name: Logistic 1
has code name: Logistic1
distribution has PDF: PDF of Logistic 1
distribution has CDF: CDF of Logistic 1
distribution has mean: mean of Logistic 1
distribution has median: median of Logistic 1
distribution has mode: mode of Logistic 1
distribution has variance: variance of Logistic 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location of Logistic-1
distribution has parameter: scale of Logistic-1

has alternate name
type: DatatypeProperty

precision of Laplace-2
type: mathematical object
has name: precision of Laplace-2
has code name: Laplace2.tau
has Latex symbol: \tau
parameter has definition expressed in plain text: \tau > 0, \tau \in R
has short name: precision
has short code name: tau

has Latex expression of support
type: DatatypeProperty

median of Log-Normal 3
type: mathematical object
has name: median of Log-Normal 3
has Latex expression: m

relationship between Binomial 1 and Bernoulli 1 whereby n=1
type: Special case
distribution relationship from: Binomial 1
distribution relationship to: Bernoulli 1
distribution relationship definition in Latex: n=1
distribution relationship pair expression Latex: Binomial1(n,p) \rightarrow Bernoulli1(p)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialBernoulli.pdf}

mean of Arcsine 1
type: mathematical object
has name: mean of Arcsine 1
has Latex expression: \frac{1}{2}

CDF of Multivariate Student T 1
type: cumulative density function
has name: CDF of Multivariate (Student) T 1
has Latex expression: \text{no analytic expression}

Generalized Gamma 1
type: probability distribution
has name: Generalized Gamma 1
has code name: GeneralizedGamma1
distribution has PDF: PDF of Generalized Gamma 1
distribution has CDF: CDF of Generalized Gamma 1
distribution has mean: mean of Generalized Gamma 1
distribution has mode: mode of Generalized Gamma 1
distribution has variance: variance of Generalized Gamma 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Generalized-Gamma-1
distribution has parameter: scale of Generalized-Gamma-1

Poisson intensity of Double-Poisson-1
type: mathematical object
has name: Poisson intensity of Double-Poisson-1
has code name: DoublePoisson1.rate
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R, \mu > 0
has short name: Poisson intensity
has short code name: rate

relationship between Log-Normal 5 and Log-Normal 3 whereby m = \\exp\\mu, \\sigma = 1 / \\sqrt{\\tau}
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 3
distribution relationship definition in Latex: m = \exp(\mu), \sigma = 1 / \sqrt{\tau}
distribution relationship pair expression Latex: LogNormal5(\mu,\tau) \rightarrow LogNormal3(m,\sigma)
distribution relationship reference: ProbOnto spec

shape of Log-Normal-2
type: mathematical object
has name: shape of Log-Normal-2
has code name: LogNormal2.varLog
has Latex symbol: v
parameter has definition expressed in plain text: v > 0
has short name: shape
has short code name: varLog

relationship between Log-Normal 7 and Log-Normal 6 whereby m = \\mu_N/\\sqrt{1+\\sigma_N^2/\\mu_N^2}, \\sigma_g = \\exp\\Big\\sqrt{\\log1+\\sigma_N^2/\\mu_N^2}\\Big
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m = \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}, \sigma_g = \exp\Big(\sqrt{\log(1+\sigma_N^2/\mu_N^2)}\Big)
distribution relationship pair expression Latex: LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal6(m,\sigma_g)
distribution relationship reference: ProbOnto spec

CDF of Standard Uniform 1
type: cumulative density function
has name: CDF of Standard Uniform 1
has Latex expression: x
has R code expression: x

has variate of mathematical type
type: DatatypeProperty

NoncentralT
type: probability distribution
has name: NoncentralT
has code name: NoncentralT1
distribution has PDF: PDF of NoncentralT
distribution has mean: mean of NoncentralT
has variate symbol in Latex: t'=\frac{x}{\sqrt{y/n}} \text{ with } x \sim Normal2(\delta,1), y \sim ChiSquared1(n).
has Latex expression of support: t' \in (0,+\infty)
distribution has parameter: degrees of freedom of NoncentralT
distribution has parameter: noncentrality parameter of NoncentralT

variance of Zero-Inflated Generalized Poisson 1
type: mathematical object
has name: variance of Zero-Inflated Generalized Poisson 1
has Latex expression: (1-p0)[\mu^2 + \mu(1 + \alpha \mu)^2] - (1-p0)^2 \mu^2

mean of Log-Normal 3
type: mathematical object
has name: mean of Log-Normal 3
has Latex expression: m e^{\frac{1}{2}\sigma^2}

shape of Two-Sided-Power-1
type: mathematical object
has name: shape of Two-Sided-Power-1
has code name: TwoSidedPower1.shape
has Latex symbol: n
parameter has definition expressed in plain text: n > 0
has short name: shape
has short code name: shape

CDF of Log-Normal 5
type: cumulative density function
has name: CDF of Log-Normal 5
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\mu}{\sqrt{2/\tau}}\Big]
has R code expression: 1/2 + 1/2 * erf( (log(x)-mu) / sqrt(2/tau) )

mean of Chi-squared 1
type: mathematical object
has name: mean of Chi-squared 1
has Latex expression: k

CDF of Bernoulli 1
type: cumulative density function
has name: CDF of Bernoulli 1
has Latex expression: \begin{cases} 0 & \text{for }k<0 \\ q & \text{for }0\leq k<1 \\ 1 & \text{for }k\geq 1 \end{cases}

coefficient of variation of Log-Normal-4
type: mathematical object
has name: coefficient of variation of Log-Normal-4
has code name: LogNormal4.coefVar
has Latex symbol: cv
parameter has definition expressed in plain text: cv>0
has short name: coefficient of variation
has short code name: coefVar

variance of Benford 1
type: mathematical object
has name: variance of Benford 1
has Latex expression: \approx 6.0565

relationship between Standard Uniform 1 and Frechet 2 whereby \\text{If } X \\sim StandardUniform10,1 \\text{ then } m+\\sigma-\\logX^{-1/\\alpha} \\sim Frechet2\\alpha, \\sigma, m
type: Transformation
distribution relationship from: Standard Uniform 1
distribution relationship to: Inverse Weibull
distribution relationship definition in Latex: \text{If } X \sim StandardUniform1(0,1) \text{ then } m+\sigma(-\log(X))^{-1/\alpha} \sim Frechet2(\alpha, \sigma, m)
distribution relationship pair expression Latex: StandardUniform1(0,1) \rightarrow Frechet2(\alpha,\sigma,m)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution}

mean of Laplace 2
type: mathematical object
has name: mean of Laplace 2
has Latex expression: \mu

HF of Erlang 1
type: hazard function
has name: HF of Erlang 1
has Latex expression: \frac{(x/b)^{c-1}}{b(c-1)! \sum^{c-1}_{i=0} \frac{(x/b)^i}{i!}}

lower triangular matrix of Multivariate-Normal-3
type: mathematical object
has name: lower triangular matrix of Multivariate-Normal-3
has code name: MultivariateNormal3.choleskyFactor
has Latex symbol: L
parameter has definition expressed in plain text: L \in R^{K \times K}, \text{ lower triangular and such that } LL^T \text{ is positive definite}, \text{with } K \in N
has short name: lower triangular matrix
has short code name: choleskyFactor

PDF of Asymmetric Laplace 1
type: probability density function
has name: PDF of Asymmetric Laplace 1
has Latex expression: \frac{1}{\sigma}\frac{\kappa}{1+\kappa^2} \begin{cases} \exp ( \frac{1}{\sigma\kappa}x ) & \text{if }x < 0 \\ \exp \left( -\frac{\kappa}{\sigma}x \right) & \text{if }x \geq 0 \end{cases}\\ \text{with } \kappa = \frac{2\sigma}{\sqrt{4\sigma^2+\mu^2}+\mu}
has R code expression: kappa = function(mu,sigma){ 2*sigma/(sqrt(4*sigma^2+mu^2)+mu) } PDF1 = function(x,mu,sigma) { 1/sigma*kappa(mu,sigma)/(1+kappa(mu,sigma)^2)*exp(1/(sigma*kappa(mu,sigma))*x ) } PDF2 = function(x,mu,sigma) { 1/sigma*kappa(mu,sigma)/(1+kappa(mu,sigma)^2)*exp( -kappa(mu,sigma)/sigma *x) }

median of Arcsine 1
type: mathematical object
has name: median of Arcsine 1
has Latex expression: \frac{1}{2}

dispersion of Double-Poisson-1
type: mathematical object
has name: dispersion of Double-Poisson-1
has code name: DoublePoisson1.dispersion
has Latex symbol: \phi
parameter has definition expressed in plain text: \phi \in R
has short name: dispersion
has short code name: dispersion

degrees of freedom of Student-s-t-distribution-1
type: mathematical object
has name: degrees of freedom of Student-s-t-distribution-1
has code name: StudentT1.degreesOfFreedom
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu > 0, \nu \in R
has short name: degrees of freedom
has short code name: degreesOfFreedom

PDF of Multivariate Student T 1
type: probability density function
has name: PDF of Multivariate (Student) T 1
has Latex expression: \frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left|\Sigma\right|^{1/2}\left[1+\frac{1}{\nu}(x-\mu)^{\rm T}\Sigma^{-1}(x-\mu)\right]^{(\nu+p)/2}}

mean of logx of Log-Normal-2
type: mathematical object
has name: mean of log(x) of Log-Normal-2
has code name: LogNormal2.meanLog
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean of log(x)
has short code name: meanLog

parameter has mathematical type
type: DatatypeProperty

mean of Zero-Inflated-Generalized-Poisson-1
type: mathematical object
has name: mean of Zero-Inflated-Generalized-Poisson-1
has code name: ZeroInflatedGeneralizedPoisson1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu > 0
has short name: mean
has short code name: mean

CDF of Half-normal 2
type: cumulative density function
has name: CDF of Half-normal 2
has Latex expression: erf\Big(\frac{x-\mu}{\sqrt{2}\sigma}\Big)
has R code expression: erf((x-mu) / (sqrt(2)*sigma))

CDF of Chi-squared 1
type: cumulative density function
has name: CDF of Chi-squared 1
has Latex expression: \frac{1}{\Gamma\left(\frac{k}{2}\right)}\; \gamma\left(\frac{k}{2},\,\frac{x}{2}\right)
has R code expression: 1/gamma(k/2) * Igamma(k/2,x/2)

PDF of Log-Normal 5
type: probability density function
has name: PDF of Log-Normal 5
has Latex expression: \sqrt{\frac{\tau}{2 \pi}} \frac{1}{x}e^{-\frac{\tau}{2}(\log x-\mu)^2}
has R code expression: sqrt(tau / (2*pi)) * (1/x) * exp(- (tau/2)*(log(x)-mu)^2 )

CDF of Log-Normal 3
type: cumulative density function
has name: CDF of Log-Normal 3
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log m}{\sqrt{2}\sigma}\Big]
has R code expression: 1/2 + 1/2 * erf( (log(x)-log(m)) / (sqrt(2)*sigma) )

location of Two-Sided-Power-1
type: mathematical object
has name: location of Two-Sided-Power-1
has code name: TwoSidedPower1.location
has Latex symbol: m
parameter has definition expressed in plain text: a < m < b
has short name: location
has short code name: location

variance of Laplace 2
type: mathematical object
has name: variance of Laplace 2
has Latex expression: 2 / \tau^2

relationship between Weibull 2 and Weibull 1 whereby k=v, \\lambda = \\lambda^{-1/v}
type: Reparameterisation
distribution relationship from: Weibull 2
distribution relationship to: Weibull 1
distribution relationship definition in Latex: k=v, \lambda = \lambda^{-1/v}
distribution relationship pair expression Latex: Weibull2(\lambda,v) \rightarrow Weibull1(\lambda,k)
distribution relationship reference: ProbOnto spec

mode of Benford 1
type: mathematical object
has name: mode of Benford 1
has Latex expression: 1

median / geometric mean of Log-Normal-4
type: mathematical object
has name: median / geometric mean of Log-Normal-4
has code name: LogNormal4.median
has Latex symbol: m
parameter has definition expressed in plain text: m>0
has short name: median / geometric mean
has short code name: median

CDF of Asymmetric Laplace 1
type: cumulative density function
has name: CDF of Asymmetric Laplace 1
has Latex expression: \begin{cases} \frac{\kappa^2}{1+\kappa^2} \exp \left( \frac{1}{\sigma\kappa}x \right) & \text{if }x \leq 0\\ 1 -\frac{1}{1+\kappa^2} \exp \left( -\frac{\kappa}{\sigma}x \right) & \text{if }x >0 \end{cases}
has R code expression: kappa = function(mu,sigma){ 2*sigma/(sqrt(4*sigma^2+mu^2)+mu) } CDF1 = function(x,mu,sigma) { kappa(mu,sigma)^2/(1+kappa(mu,sigma)^2)*exp(1/(sigma*kappa(mu,sigma))*x )} CDF2 = function(x,mu,sigma) { 1-1/(1+kappa(mu,sigma)^2)*exp(-kappa(mu,sigma)/sigma*x)}

location of Multivariate-Normal-3
type: mathematical object
has name: location of Multivariate-Normal-3
has code name: MultivariateNormal3.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R^K
has short name: location
has short code name: mean

SF of Erlang 1
type: survival function
has name: SF of Erlang 1
has Latex expression: \Big[\exp\Big(-\frac{x}{b}\Big)\Big]\Big( \sum^{c-1}_{i=0} \frac{(x/b)^i}{i!}\Big)

Multivariate Student T 1
type: probability distribution
has name: Multivariate (Student) T 1
has code name: MultivariateStudentT1
distribution has PDF: PDF of Multivariate Student T 1
distribution has CDF: CDF of Multivariate Student T 1
distribution has mean: mean of Multivariate Student T 1
distribution has median: median of Multivariate Student T 1
distribution has mode: mode of Multivariate Student T 1
distribution has variance: variance of Multivariate Student T 1
has variate symbol in Latex: x
has Latex expression of support: x \in R^p
distribution has parameter: covariance matrix of Multivariate-Student-T-1
distribution has parameter: location of Multivariate-Student-T-1
distribution has parameter: degrees of freedom of Multivariate-Student-T-1

mode of Categorical Ordered 1
type: mathematical object
has name: mode of Categorical Ordered 1
has Latex expression: i\text{ such that }p_i=\max(p_1, \ldots, p_k)

CDF of Generalized Gamma 1
type: cumulative density function
has name: CDF of Generalized Gamma 1
has Latex expression: \frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)}
has R code expression: Igamma(d/p, (x/a)^p, lower=T) / gamma(d/p)

mean of Double Poisson 1
type: mathematical object
has name: mean of Double Poisson 1
has Latex expression: \mu

Standard Uniform 1
type: probability distribution
label: Rectangular
has name: Standard Uniform 1
has code name: StandardUniform1
distribution has PDF: PDF of Standard Uniform 1
distribution has CDF: CDF of Standard Uniform 1
distribution has SF: SF of Standard Uniform 1
distribution has mean: mean of Standard Uniform 1
distribution has median: median of Standard Uniform 1
distribution has mode: mode of Standard Uniform 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,1]
distribution has parameter: minimum of Standard-Uniform-1
distribution has parameter: maximum of Standard-Uniform-1

PMF of Zero-Inflated Generalized Poisson 1
type: probability mass function
has name: PMF of Zero-Inflated Generalized Poisson 1
has Latex expression: \begin{cases} p0 + (1-p0) \exp\Big[ \frac{-\mu}{1+\alpha \mu}\Big] & \text{for } y = 0 \\ (1-p0) \Big( \frac{\mu}{1+\alpha \mu}\Big)^y \frac{(1+\alpha y)^{y-1}}{y!} \exp\Big[ \frac{-\mu (1+\alpha y)}{1+\alpha \mu}\Big] & \text{for } y > 0 \end{cases}
has R code expression: PMF1=p0 + (1-p0) * exp(-mu/(1+ alpha*mu)) # for y = 0 PMF2=(1-p0)*( mu/(1+alpha*mu))^y*((1+alpha*y)^(y-1))/factorial(y)*exp((-mu*(1+alpha*y))/(1+alpha*mu)) # for y > 0

has variate symbol in Latex
type: DatatypeProperty

Categorical Nonordered 1
type: probability distribution
has name: Categorical Nonordered 1
has code name: CategoricalNonordered1
distribution has PMF: PMF of Categorical Nonordered 1
distribution has CDF: CDF of Categorical Nonordered 1
distribution has mean: mean of Categorical Nonordered 1
distribution has median: median of Categorical Nonordered 1
distribution has mode: mode of Categorical Nonordered 1
distribution has variance: variance of Categorical Nonordered 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{1,\dots,k\}
distribution has parameter: category probabilities of Categorical-Nonordered-1

relationship between Rice 1 and Rayleigh 1 whereby \\text{If } R \\sim Rice10,\\sigma \\text{ then } R \\sim Rayleigh1\\sigma
type: Special case
distribution relationship from: Rician
distribution relationship to: Rayleigh 1
distribution relationship definition in Latex: \text{If } R \sim Rice1(0,\sigma) \text{ then } R \sim Rayleigh1(\sigma)
distribution relationship pair expression Latex: Rice1(\nu,\sigma) \rightarrow Rayleigh1(\sigma)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Rice_distribution}

relationship between Zero-Inflated Generalized Poisson 1 and Poisson 1 whereby p0=0, \\alpha=0, \\lambda=\\mu
type: Reparameterisation
distribution relationship from: Zero-Inflated Generalized Poisson 1
distribution relationship to: Poisson 1
distribution relationship definition in Latex: p0=0, \alpha=0, \lambda=\mu
distribution relationship pair expression Latex: ZeroInflatedGeneralizedPoisson1(\mu,\alpha,p0) \rightarrow Poisson1(\lambda)
distribution relationship reference: ProbOnto spec

median of Benford 1
type: mathematical object
has name: median of Benford 1
has Latex expression: 3

mode of Chi-squared 1
type: mathematical object
has name: mode of Chi-squared 1
has Latex expression: max\{k-2,0\}

PDF of Laplace 2
type: probability density function
has name: PDF of Laplace 2
has Latex expression: \frac{\tau}{2} \exp \left(-\tau|x-\mu| \right)
has R code expression: tau/2 * exp(-tau * abs(x-mu))

SF of Weibull 2
type: survival function
has name: SF of Weibull 2
has Latex expression: \exp(-x^v \lambda)
has R code expression: exp(-x^v * lambda)

mean of Normal 2
type: mathematical object
has name: mean of Normal 2
has Latex expression: \mu

scale parameter of Burr-1
type: mathematical object
has name: scale parameter of Burr-1
has code name: Burr1.scale
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: scale parameter
has short code name: scale

scale of Half-normal-2
type: mathematical object
has name: scale of Half-normal-2
has code name: HalfNormal2.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: scale
has short code name: scale

PDF of Erlang 1
type: probability density function
has name: PDF of Erlang 1
has Latex expression: \frac{(x/b)^{c-1} \exp(-x/b)}{b(c-1)!}
has R code expression: ((x/b)^(c-1) *exp(-x/b)) / (b * factorial(c-1))

shape parameter of Weibull-Discrete-1
type: mathematical object
has name: shape parameter of Weibull-Discrete-1
has code name: WeibullDiscrete1.shape1
has Latex symbol: p
parameter has definition expressed in plain text: 0<p<1
has short name: shape parameter
has short code name: shape1

variance of Multivariate Normal 3
type: mathematical object
has name: variance of Multivariate Normal 3
has Latex expression: LL^T

variance of Double Poisson 1
type: mathematical object
has name: variance of Double Poisson 1
has Latex expression: \mu/\phi

median of Categorical Ordered 1
type: mathematical object
has name: median of Categorical Ordered 1
has Latex expression: i\text{ such that }\sum_{j=1}^{i-1} p_j \leq 0.5\text{ and }\sum_{j=1}^{i} p_j \geq 0.5

PDF of Generalized Gamma 1
type: probability density function
has name: PDF of Generalized Gamma 1
has Latex expression: \frac{p/a^d}{\Gamma(d/p)} x^{d-1}e^{-(x/a)^p}
has R code expression: p/a^d/gamma(d/p) * x^(d-1) * exp(-(x/a)^p)

SF of Arcsine 1
type: survival function
has name: SF of Arcsine 1
has Latex expression: \frac{\pi -2 \arcsin\left(2x -1\right)}{2\pi}
has R code expression: (pi - 2 * asin(2*x -1)) / (2*pi)

has variate of mathematical type
type: ObjectProperty

CDF of Zero-Inflated Generalized Poisson 1
type: cumulative density function
has name: CDF of Zero-Inflated Generalized Poisson 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: c(PMF1,cumsum(PMF2)+PMF1)

mean of Log-Normal 5
type: mathematical object
has name: mean of Log-Normal 5
has Latex expression: e^{\mu + \frac{1}{2\tau}}

PDF of Standard Uniform 1
type: probability density function
has name: PDF of Standard Uniform 1
has Latex expression: 1
has R code expression: 1

PMF of Categorical Nonordered 1
type: probability mass function
has name: PMF of Categorical Nonordered 1
has Latex expression: p(x=i)=p_i

relationship between Log-Normal 1 and Normal 1 whereby \\logX
type: Transformation
distribution relationship from: Log-Normal 1
distribution relationship to: Normal
distribution relationship definition in Latex: \log(X)
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalLognormal.pdf}

Scaled Inverse Chi-Square
type: probability distribution
has name: Scaled Inverse Chi-Square
has code name: ScaledInverseChiSquare1
distribution has PDF: PDF of Scaled Inverse Chi-Square
distribution has CDF: CDF of Scaled Inverse Chi-Square
distribution has mean: mean of Scaled Inverse Chi-Square
distribution has mode: mode of Scaled Inverse Chi-Square
distribution has variance: variance of Scaled Inverse Chi-Square
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: degrees of freedom of Scaled-Inverse-Chi-Square
distribution has parameter: scaling parameter of Scaled-Inverse-Chi-Square

PROB k0000948
type: Transformation
label: relationship between Exponential 2 and F 1 whereby \\text{If } X_1, X_2 \\sim Exponential21 \\text{ mutually independent and identically distributed} \\\\ \\text{ random variables } \\Rightarrow X_1/X_2 \\text{ has the } F1 \\text{ distribution}
distribution relationship from: Negative exponential
distribution relationship to: F 1
distribution relationship definition in Latex: \text{If } X_1, X_2 \sim Exponential2(1) \text{ mutually independent and identically distributed} \\ \text{ random variables } \Rightarrow X_1/X_2 \text{ has the } F1 \text{ distribution}
distribution relationship pair expression Latex: Exponential2(1) \rightarrow F1(n_1,n_2)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ExponentialF.pdf}

median of Normal 2
type: mathematical object
has name: median of Normal 2
has Latex expression: \mu

HF of Log-Logistic 2
type: hazard function
has name: HF of Log-Logistic 2
has Latex expression: \frac{\lambda \kappa (\lambda x)^{\kappa-1}}{1+(\lambda x)^\kappa}
has R code expression: lambda * kappa * (lambda * x)^(kappa-1) / (1+(lambda * x)^kappa)

mean of Zero-Inflated Generalized Poisson 1
type: mathematical object
has name: mean of Zero-Inflated Generalized Poisson 1
has Latex expression: (1-p0)\mu

CDF of Laplace 2
type: cumulative density function
has name: CDF of Laplace 2
has Latex expression: \begin{cases} \frac12 \exp \left( \tau (x-\mu) \right) & \mbox{if }x < \mu \\ 1-\frac12 \exp \left( -\tau (x-\mu) \right) & \mbox{if }x \geq \mu \end{cases}
has R code expression: 1/2 * exp( tau*(x-mu) ) # for x < mu 1- 1/2 * exp( -tau*(x-mu) ) # x >= mu

median of Chi-squared 1
type: mathematical object
has name: median of Chi-squared 1
has Latex expression: \approx k\bigg(1-\frac{2}{9k}\bigg)^3

SF of Burr 1
type: survival function
has name: SF of Burr 1
has Latex expression: \frac{1}{\Big[1+(x/\alpha)^c\Big]^k}
has R code expression: 1/(1+(x/alpha)^c)^k

Asymmetric Laplace 1
type: probability distribution
has name: Asymmetric Laplace 1
has code name: AsymmetricLaplace1
distribution has PDF: PDF of Asymmetric Laplace 1
distribution has CDF: CDF of Asymmetric Laplace 1
distribution has mean: mean of Asymmetric Laplace 1
distribution has variance: variance of Asymmetric Laplace 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location parameter of Asymmetric-Laplace-1
distribution has parameter: scale parameter of Asymmetric-Laplace-1

CDF of Zero-inflated Poisson 1
type: cumulative density function
has name: CDF of Zero-inflated Poisson 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: c(PMF1,cumsum(PMF2)+PMF1)

SF of Weibull Discrete 1
type: survival function
has name: SF of Weibull Discrete 1
has Latex expression: (1-p)^{x^\beta}
has R code expression: (1-p)^(x^beta)

location of Half-normal-2
type: mathematical object
has name: location of Half-normal-2
has code name: HalfNormal2.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

CDF of Erlang 1
type: cumulative density function
has name: CDF of Erlang 1
has Latex expression: 1 - \Big[\exp\Big(-\frac{x}{b}\Big)\Big]\Big( \sum^{c-1}_{i=0} \frac{(x/b)^i}{i!}\Big)
has R code expression: R function

relationship between Beta 1 and Arcsine 1 whereby \\alpha=1/2, \\beta = 1/2
type: Special case
distribution relationship from: Beta 1
distribution relationship to: Arcsine 1
distribution relationship definition in Latex: \alpha=1/2, \beta = 1/2
distribution relationship pair expression Latex: Beta1(\alpha,\beta) \rightarrow Arcsine1
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/UDR.html}

median of Conway-Maxwell-Poisson 1
type: mathematical object
has name: median of Conway-Maxwell-Poisson 1
has Latex expression: \text{No closed form}

mean of Half-normal 2
type: mathematical object
has name: mean of Half-normal 2
has Latex expression: \mu + \sigma \sqrt{\frac{2}{\pi}}

non denumerable set
type: Class
subclass of: mathematical set

shape parameter of Weibull-Discrete-1
type: mathematical object
has name: shape parameter of Weibull-Discrete-1
has code name: WeibullDiscrete1.shape2
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape parameter
has short code name: shape2

Lomax 1
type: probability distribution
has name: Lomax 1
has code name: Lomax1
distribution has PDF: PDF of Lomax 1
distribution has CDF: CDF of Lomax 1
distribution has mean: mean of Lomax 1
distribution has median: median of Lomax 1
distribution has mode: mode of Lomax 1
distribution has variance: variance of Lomax 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: shape of Lomax-1
distribution has parameter: scale of Lomax-1

relationship between Two-Sided Power 1 and Uniform 1 whereby n=1
type: Special case
distribution relationship from: Two-Sided Power 1
distribution relationship to: Uniform 1
distribution relationship definition in Latex: n=1
distribution relationship pair expression Latex: TwoSidedPower1(a,b,m,n) \rightarrow Uniform1(a,b)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TSPUniform.pdf}

variance of Frechet 1
type: mathematical object
has name: variance of Frechet 1
has Latex expression: \begin{cases} \ \sigma^2\left(\Gamma\left(1-\frac{2}{\alpha}\right)- \left(\Gamma\left(1-\frac{1}{\alpha}\right)\right)^2\right) & \text{for } \alpha>2 \\ \ \infty & \text{otherwise} \end{cases}

distribution relationship definition in Latex
type: DatatypeProperty

mode of Cauchy 1
type: mathematical object
has name: mode of Cauchy 1
has Latex expression: x_0

scale of Exponential-2
type: mathematical object
has name: scale of Exponential-2
has code name: Exponential2.mean
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: scale
has short code name: mean

minimum of Uniform-1
type: mathematical object
has name: minimum of Uniform-1
has code name: Uniform1.minimum
has Latex symbol: a
parameter has definition expressed in plain text: a \in R
has short name: minimum
has short code name: minimum

relationship between Pareto Type I and Exponential 1 whereby X \\sim ParetoTypeI1,Y= \\logX/\\lambda \\Rightarrow Y \\sim Exponential1
type: Transformation
distribution relationship from: Pareto Type I
distribution relationship to: Exponential 1
distribution relationship definition in Latex: X \sim ParetoTypeI1,Y= \log(X/\lambda) \Rightarrow Y \sim Exponential1
distribution relationship pair expression Latex: ParetoTypeI1(x_m,\alpha) \rightarrow Exponential1(\lambda)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ParetoExponential.pdf}

median of Levy 1
type: mathematical object
has name: median of Levy 1
has Latex expression: c/2(\textrm{erfc}^{-1}(1/2))^2 \text{ for } \mu=0

has variate symbol
type: DatatypeProperty

scale of Pareto-Type-I
type: mathematical object
has name: scale of Pareto-Type-I
has code name: ParetoTypeI1.scale
has Latex symbol: x_m
parameter has definition expressed in plain text: x_m > 0, x_m \in R
has short name: scale
has short code name: scale

CDF of Standard Normal 1
type: cumulative density function
has name: CDF of Standard Normal 1
has Latex expression: \frac12\left[1 + \text{erf}\left( \frac{x}{\sqrt{2}}\right)\right]
has R code expression: 1/2 * (1 + erf(x/(sqrt(2))))

PROB k0001406
type: Special case
label: relationship between Amoroso 1 and Rayleigh 1 whereby \\text{Rayleigh1}\\sigma \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a = 0, \\theta=\\sqrt{2\\sigma^2}, \\alpha=1, \\beta=2
distribution relationship from: Amoroso 1
distribution relationship to: Rayleigh 1
distribution relationship definition in Latex: \text{Rayleigh1}(\sigma) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a = 0, \theta=\sqrt{2\sigma^2}, \alpha=1, \beta=2
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Rayleigh1(\sigma)
distribution relationship reference: \cite{crooks2010amoroso}

PDF of Multivariate Gaussian Process Distribution 1
type: probability density function
has name: PDF of Multivariate Gaussian Process Distribution 1
has Latex expression: \prod^K_{i=1} \text{MultivariateNormal1}(x_i|0,w^{-1}_i \Sigma)

relationship between Bernoulli 2 and Bernoulli 1 whereby p = \\exp\\alpha / 1 + \\exp\\alpha
type: Reparameterisation
distribution relationship from: Bernoulli 2
distribution relationship to: Bernoulli 1
distribution relationship definition in Latex: p = \exp(\alpha) / (1 + \exp(\alpha))
distribution relationship pair expression Latex: Bernoulli2(\alpha) \rightarrow Bernoulli1(p)
distribution relationship reference: ProbOnto spec

mean of Generalized Gamma 1
type: mathematical object
has name: mean of Generalized Gamma 1
has Latex expression: a \frac{\Gamma((d+1)/p)}{\Gamma(d/p)}

mode of Log-Normal 7
type: mathematical object
has name: mode of Log-Normal 7
has Latex expression: \frac{\mu_N}{(1+\sigma_N^2/\mu_N^2)^{3/2}}

mean of Categorical Nonordered 1
type: mathematical object
has name: mean of Categorical Nonordered 1
has Latex expression: E([x=i]) = p_i, \text{this is the mean of the Iverson bracket } [x=i] \text{ and not the mean of } x

has Latex symbol
type: DatatypeProperty

relationship between Pareto Type II and Lomax 1 whereby \\mu=0
type: Reparameterisation
distribution relationship from: Pareto Type II
distribution relationship to: Lomax 1
distribution relationship definition in Latex: \mu=0
distribution relationship pair expression Latex: ParetoTypeII1(\mu,\lambda,\alpha) \rightarrow Lomax1(\lambda,\alpha)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I.E2.80.93IV}

PROB k0001446
type: Limiting
label: relationship between Amoroso 1 and Normal 1 whereby \\text{Normal1}\\mu,\\sigma \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a=\\mu-\\sigma\\sqrt{\\alpha}, \\theta=\\sigma/\\sqrt{\\alpha}, \\beta=1 \\text{ and } \\alpha \\rightarrow \\infty
distribution relationship from: Amoroso 1
distribution relationship to: Normal
distribution relationship definition in Latex: \text{Normal1}(\mu,\sigma) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a=\mu-\sigma\sqrt{\alpha}, \theta=\sigma/\sqrt{\alpha}, \beta=1 \text{ and } \alpha \rightarrow \infty
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{crooks2010amoroso}

variance of Half-normal 2
type: mathematical object
has name: variance of Half-normal 2
has Latex expression: \sigma^2 (1-\frac{2}{\pi})

variance of Conway-Maxwell-Poisson 1
type: mathematical object
has name: variance of Conway-Maxwell-Poisson 1
has Latex expression: \sum_{j=0}^\infty \frac{j^2\lambda^j}{(j!)^\nu Z(\lambda, \nu)} - \text{Mean}^2

median of Cauchy 1
type: mathematical object
has name: median of Cauchy 1
has Latex expression: x_0

distribution relationship to
type: ObjectProperty

PDF of Lomax 1
type: probability density function
has name: PDF of Lomax 1
has Latex expression: {\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}
has R code expression: alpha/lambda* (1+ x/lambda)^(-(alpha+1))

Borel 1
type: probability distribution
has name: Borel 1
has code name: Borel1
distribution has PMF: PMF of Borel 1
distribution has CDF: CDF of Borel 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{1,2,\dots,\infty\}
distribution has parameter: rate of Borel-1

denumerable set
type: Class
subclass of: mathematical set

shape of Frechet-1
type: mathematical object
has name: shape of Frechet-1
has code name: Frechet1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R^+
has short name: shape
has short code name: shape

relationship between Two-Sided Power 1 and Triangular 1 whereby n=2
type: Special case
distribution relationship from: Two-Sided Power 1
distribution relationship to: Triangular 1
distribution relationship definition in Latex: n=2
distribution relationship pair expression Latex: TwoSidedPower1(a,b,m,n) \rightarrow Triangular1(a,b,m)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TSPTriangular.pdf}

Log-Normal 2
type: probability distribution
label: lognormal
label: Galton
has name: Log-Normal 2
has code name: LogNormal2
distribution has PDF: PDF of Log-Normal 2
distribution has CDF: CDF of Log-Normal 2
distribution has mean: mean of Log-Normal 2
distribution has median: median of Log-Normal 2
distribution has mode: mode of Log-Normal 2
distribution has variance: variance of Log-Normal 2
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: mean of logx of Log-Normal-2
distribution has parameter: shape of Log-Normal-2

has type name
type: DatatypeProperty

variance of Pareto Type I
type: mathematical object
has name: variance of Pareto Type I
has Latex expression: \begin{cases} \infty & \text{for }\alpha\in(1,2] \\ \frac{x_m^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 \end{cases}

PDF of Standard Normal 1
type: probability density function
has name: PDF of Standard Normal 1
has Latex expression: \frac{e^{-\frac{1}{2} x^2}}{\sqrt{2\pi}}
has R code expression: 1/(sqrt(2*pi))*exp(-x^2/2)

relationship between Sinh-Arcsinh 2 and Normal 1 whereby \\epsilon=0,\\delta=1
type: Special case
distribution relationship from: Sinh-Arcsinh 2
distribution relationship to: Normal
distribution relationship definition in Latex: \epsilon=0,\delta=1
distribution relationship pair expression Latex: SinhArcsinh2(\mu,\sigma,\epsilon,\delta) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{rubio2013modelling}

mode of Levy 1
type: mathematical object
has name: mode of Levy 1
has Latex expression: \frac{c}{3} \text{ for } \mu=0

PMF of Borel 1
type: probability mass function
has name: PMF of Borel 1
has Latex expression: \frac{e^{\mu n} (\mu n)^{n-1}}{n!}
has R code expression: e^(mu*n)*(mu*n)^(n-1) / factorial(n)

Multivariate Gaussian Process Distribution 1
type: probability distribution
has name: Multivariate Gaussian Process Distribution 1
has code name: MultivariateGaussianProcess1
distribution has PDF: PDF of Multivariate Gaussian Process Distribution 1
has variate symbol in Latex: x
has Latex expression of support: x \in R^{K\times N}
distribution has parameter: vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-1
distribution has parameter: symmetric, positive definite kernel matrix of Multivariate-Gaussian-Process-Distribution-1

relationship between Truncated Normal 1 and Half-normal 1 whereby \\mu=0, a=0, b=\\infty
type: Special case
distribution relationship from: Truncated Normal 1
distribution relationship to: Half-normal 1
distribution relationship definition in Latex: \mu=0, a=0, b=\infty
distribution relationship pair expression Latex: TruncatedNormal1(\mu,\sigma,a,b) \rightarrow HalfNormal1(\theta)
distribution relationship reference: \url{http://reference.wolfram.com/language/ref/HalfNormalDistribution.html} \\ \cite{forbes2011statistical}

has comment
type: DatatypeProperty

relationship between Negative Binomial 3 and Negative Binomial 1 whereby r=\\phi, p=r/\\mu+r
type: Reparameterisation
distribution relationship from: Negative Binomial 3
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: r=\phi, p=r/(\mu+r)
distribution relationship pair expression Latex: NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial1(r, p)
distribution relationship reference: ProbOnto spec

CDF of Categorical Nonordered 1
type: cumulative density function
has name: CDF of Categorical Nonordered 1
has Latex expression: undefined

mode of Generalized Gamma 1
type: mathematical object
has name: mode of Generalized Gamma 1
has Latex expression: a \left(\frac{d-1}{p}\right)^{\frac{1}{p}}, \text{ for } d>1

median of Log-Normal 7
type: mathematical object
has name: median of Log-Normal 7
has Latex expression: \frac{\mu_N}{\sqrt{1+\sigma_N^2/\mu_N^2}}

CDF of Conway-Maxwell-Poisson 1
type: cumulative density function
has name: CDF of Conway-Maxwell-Poisson 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

Log-Normal 3
type: probability distribution
label: lognormal
label: Galton
has name: Log-Normal 3
has code name: LogNormal3
distribution has PDF: PDF of Log-Normal 3
distribution has CDF: CDF of Log-Normal 3
distribution has mean: mean of Log-Normal 3
distribution has median: median of Log-Normal 3
distribution has mode: mode of Log-Normal 3
distribution has variance: variance of Log-Normal 3
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: median / geometric mean of Log-Normal-3
distribution has parameter: shape of Log-Normal-3

relationship between Arcsine 2 and Arcsine 1 whereby a=0, b=1
type: Special case
distribution relationship from: Arcsine 2
distribution relationship to: Arcsine 1
distribution relationship definition in Latex: a=0, b=1
distribution relationship pair expression Latex: Arcsine2(a,b) \rightarrow Arcsine1
distribution relationship reference: ProbOnto spec

distribution relationship from
type: ObjectProperty

enumeration
type: Class
subclass of: mathematical set

PDF of Half-normal 2
type: probability density function
has name: PDF of Half-normal 2
has Latex expression: \sqrt{\frac{2}{\pi}} \frac{1}{\sigma} \exp\Big[-\frac{1}{2}\Big(\frac{x-\mu}{\sigma}\Big)^2\Big]
has R code expression: sqrt(2/pi)*1/sigma * exp(-1/2 * ((x-mu)/sigma)^2)

relationship between Log-Normal 2 and Log-Normal 6 whereby m=\\exp\\mu, \\sigma_g=\\exp\\sqrt{v}
type: Reparameterisation
distribution relationship from: Log-Normal 2
distribution relationship to: lognormal
distribution relationship definition in Latex: m=\exp(\mu), \sigma_g=\exp(\sqrt{v})
distribution relationship pair expression Latex: LogNormal2(\mu,v) \rightarrow LogNormal6(m,\sigma_g)
distribution relationship reference: ProbOnto spec

mean of Cauchy 1
type: mathematical object
has name: mean of Cauchy 1
has Latex expression: undefined

scale of Frechet-1
type: mathematical object
has name: scale of Frechet-1
has code name: Frechet1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma \in R^+
has short name: scale
has short code name: scale

relationship between Standard Two-Sided Power 1 and Standard Uniform 1 whereby n=1
type: Special case
distribution relationship from: Standard Two-Sided Power 1
distribution relationship to: Standard Uniform 1
distribution relationship definition in Latex: n=1
distribution relationship pair expression Latex: StandardTwoSidedPower1(m,n) \rightarrow StandardUniform1

PROB k0001408
type: Special case
label: relationship between Amoroso 1 and Inverse Chi-Square whereby \\text{InverseChiSquare1}k \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } a = 0, \\theta=1/2, \\alpha=k/2, \\beta=-1
distribution relationship from: Amoroso 1
distribution relationship to: Inverted-chi-square
distribution relationship definition in Latex: \text{InverseChiSquare1}(k) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } a = 0, \theta=1/2, \alpha=k/2, \beta=-1
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow InverseChiSquare1(k)
distribution relationship reference: \cite{crooks2010amoroso}

variance of Levy 1
type: mathematical object
has name: variance of Levy 1
has Latex expression: \infty

Generalized Poisson 1
type: probability distribution
has name: Generalized Poisson 1
has code name: GeneralizedPoisson1
distribution has PMF: PMF of Generalized Poisson 1
distribution has CDF: CDF of Generalized Poisson 1
distribution has mean: mean of Generalized Poisson 1
distribution has variance: variance of Generalized Poisson 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: dispersion of Generalized-Poisson-1
distribution has parameter: Poisson intensity of Generalized-Poisson-1

has short name
type: DatatypeProperty

mode of Exponential 2
type: mathematical object
has name: mode of Exponential 2
has Latex expression: 0

PROB k0001072
type: Reparameterisation
label: relationship between Log-Normal 6 and Log-Normal 7 whereby \\mu_N = m \\exp\\big\\frac12 \\log^2\\sigma_g\\big, \\sigma_N = m \\exp\\big\\frac12 \\log^2\\sigma_g\\big \\sqrt{\\exp\\big[\\log^2\\sigma_g\\big]-1 }
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu_N = m \exp\big(\frac12 \log^2(\sigma_g)\big), \sigma_N = m \exp\big(\frac12 \log^2(\sigma_g)\big) \sqrt{\exp\big[\log^2(\sigma_g)\big]-1 }
distribution relationship pair expression Latex: LogNormal6(m,\sigma_g) \rightarrow LogNormal7(\mu_N,\sigma_N)
distribution relationship reference: ProbOnto spec

mean of Log-Normal 7
type: mathematical object
has name: mean of Log-Normal 7
has Latex expression: \mu_N

symmetric, positive definite kernel matrix of Multivariate-Gaussian-Process-Distribution-1
type: mathematical object
has name: symmetric, positive definite kernel matrix of Multivariate-Gaussian-Process-Distribution-1
has code name: MultivariateGaussianProcess1.kernelMatrix
has Latex symbol: \Sigma
parameter has definition expressed in plain text: \Sigma \in R^{N \times N}, N \in N
has short name: symmetric, positive definite kernel matrix
has short code name: kernelMatrix

variance of Generalized Gamma 1
type: mathematical object
has name: variance of Generalized Gamma 1
has Latex expression: a^2\left(\frac{\Gamma((d+2)/p)}{\Gamma(d/p)} - \left(\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}\right)^2\right)

location of Generalized-Gamma-2
type: mathematical object
has name: location of Generalized-Gamma-2
has code name: GeneralizedGamma2.location
has Latex symbol: a
parameter has definition expressed in plain text: a > 0
has short name: location
has short code name: location

median of Standard Normal 1
type: mathematical object
has name: median of Standard Normal 1
has Latex expression: 0

mode of Pareto Type I
type: mathematical object
has name: mode of Pareto Type I
has Latex expression: x_m

CDF of Lomax 1
type: cumulative density function
has name: CDF of Lomax 1
has Latex expression: 1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}
has R code expression: 1- (1+ x/lambda)^(-alpha)

CDF of Standard Triangular 1
type: cumulative density function
has name: CDF of Standard Triangular 1
has Latex expression: \begin{cases}\frac12 x^2+x+\frac12 & \text{ for } -1 < x < 0\\ -\frac12 x^2+x+\frac12 & \text{ for } 0 \leq x < 1\end{cases}
has R code expression: CDF1=1/2*x^2+x+1/2 CDF2=-1/2*x^2+x+1/2

relationship between Standard Uniform 1 and Uniform 1 whereby X \\sim StandardNormal1 \\text{ and } Y = a + b-a X \\Rightarrow Y \\sim Uniform1
type: Transformation
distribution relationship from: Standard Uniform 1
distribution relationship to: Uniform 1
distribution relationship definition in Latex: X \sim StandardNormal1 \text{ and } Y = a + (b-a) X \Rightarrow Y \sim Uniform1
distribution relationship pair expression Latex: StandardUniform1(0,1) \rightarrow Uniform1(a,b)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformUniform.pdf}

CDF of Borel 1
type: cumulative density function
has name: CDF of Borel 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

has MathML symbol
type: DatatypeProperty

Standard Triangular 1
type: probability distribution
has name: Standard Triangular 1
has code name: StandardTriangular1
distribution has PDF: PDF of Standard Triangular 1
distribution has CDF: CDF of Standard Triangular 1
distribution has SF: SF of Standard Triangular 1
distribution has HF: HF of Standard Triangular 1
distribution has mean: mean of Standard Triangular 1
distribution has median: median of Standard Triangular 1
distribution has mode: mode of Standard Triangular 1
distribution has variance: variance of Standard Triangular 1
has variate symbol in Latex: x
has Latex expression of support: 0 \leq x \leq 1

PDF of Log-Normal 3
type: probability density function
has name: PDF of Log-Normal 3
has Latex expression: \frac{1}{x\sigma\sqrt{2\pi}}\ e^{-\frac{\left[\log (x/m)\right]^2}{2\sigma^2}}
has R code expression: 1/(x*sigma*sqrt(2*pi)) * exp(-(log(x/m))^2 / (2*sigma^2))

mean of Conway-Maxwell-Poisson 1
type: mathematical object
has name: mean of Conway-Maxwell-Poisson 1
has Latex expression: \sum_{j=0}^\infty \frac{j\lambda^j}{(j!)^\nu Z(\lambda, \nu)}

CDF of Cauchy 1
type: cumulative density function
has name: CDF of Cauchy 1
has Latex expression: \frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}
has R code expression: 1/pi * atan((x-x0)/gamma)+1/2

Model formulation
type: Class
subclass of: relationship between distributions

infinite set
type: Class
subclass of: mathematical set

PROB k0001448
type: Special case
label: relationship between Amoroso 1 and Half-normal 2 whereby \\text{HalfNormal2}\\mu,\\sigma \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution for } \\mu=0 \\text{ and when } a = 0, \\theta=\\sqrt{2\\sigma^2}, \\alpha=1/2, \\beta=2
distribution relationship from: Amoroso 1
distribution relationship to: Half-normal 2
distribution relationship definition in Latex: \text{HalfNormal2}(\mu,\sigma) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution for } \mu=0 \text{ and when } a = 0, \theta=\sqrt{2\sigma^2}, \alpha=1/2, \beta=2
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow HalfNormal2(\mu,\sigma)
distribution relationship reference: \cite{crooks2010amoroso}

relationship between Log-Normal 3 and Log-Normal 1 whereby \\mu = \\logm, \\sigma = \\sigma
type: Reparameterisation
distribution relationship from: Log-Normal 3
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \mu = \log(m), \sigma = \sigma
distribution relationship pair expression Latex: LogNormal3(m,\sigma) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

shape parameter of Noncentral-Beta-1
type: mathematical object
has name: shape parameter of Noncentral-Beta-1
has code name: NoncentralBeta1.shape2
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape parameter
has short code name: shape2

lognormal
type: probability distribution
label: Log-Normal 5
label: Galton
has name: Log-Normal 5
has code name: LogNormal5
distribution has PDF: PDF of Log-Normal 5
distribution has CDF: CDF of Log-Normal 5
distribution has mean: mean of Log-Normal 5
distribution has median: median of Log-Normal 5
distribution has mode: mode of Log-Normal 5
distribution has variance: variance of Log-Normal 5
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: mean of logx of Log-Normal-5
distribution has parameter: precision of Log-Normal-5

relationship between Normal 1 and Levy 1 whereby Y\\,\\sim\\,\\textrm{Normal}\\mu,\\sigma \\Rightarrow {Y-\\mu}^{-2} \\sim\\,\\textrm{Levy}0,1/\\sigma
type: Transformation
distribution relationship from: Normal
distribution relationship to: Levy 1
distribution relationship definition in Latex: Y\,\sim\,\textrm{Normal}(\mu,\sigma) \Rightarrow {(Y-\mu)}^{-2} \sim\,\textrm{Levy}(0,1/\sigma)
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow Levy1(\mu,c)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/L%C3%A9vy_distribution#Related_distributions}

relationship between Sinh-Arcsinh 2 and Standard Normal 1 whereby \\mu=0, \\sigma=1, \\epsilon=0,\\delta=1
type: Special case
distribution relationship from: Sinh-Arcsinh 2
distribution relationship to: Standard Normal 1
distribution relationship definition in Latex: \mu=0, \sigma=1, \epsilon=0,\delta=1
distribution relationship pair expression Latex: SinhArcsinh2(\mu,\sigma,\epsilon,\delta) \rightarrow StandardNormal1
distribution relationship reference: \cite{rubio2013modelling}

relationship between Log-Normal 7 and Log-Normal 5 whereby m = \\log \\Big\\mu_N/\\sqrt{1+\\sigma_N^2/\\mu_N^2}\\Big, \\tau = 1/\\log\\big1+\\sigma_N^2/\\mu_N^2\\big
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: m = \log \Big(\mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}\Big), \tau = 1/\log\big(1+\sigma_N^2/\mu_N^2\big)
distribution relationship pair expression Latex: LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal5(\mu,\tau)
distribution relationship reference: ProbOnto spec

variance of Exponential 2
type: mathematical object
has name: variance of Exponential 2
has Latex expression: \beta^2

maximum of Uniform-1
type: mathematical object
has name: maximum of Uniform-1
has code name: Uniform1.maximum
has Latex symbol: b
parameter has definition expressed in plain text: b \in R, a < b
has short name: maximum
has short code name: maximum

scale of Generalized-Gamma-1
type: mathematical object
has name: scale of Generalized-Gamma-1
has code name: GeneralizedGamma1.scale
has Latex symbol: a
parameter has definition expressed in plain text: a > 0
has short name: scale
has short code name: scale

has plain text symbol
type: DatatypeProperty

scale of Generalized-Gamma-2
type: mathematical object
has name: scale of Generalized-Gamma-2
has code name: GeneralizedGamma2.scale
has Latex symbol: b
parameter has definition expressed in plain text: b > 0
has short name: scale
has short code name: scale

PROB k0000905
type: Transformation
label: relationship between Standard Uniform 1 and Beta 1 whereby aX_1, X_2,... , X_n iid \\sim StandardNormal1 \\Rightarrow \\text{ with } \\alpha=r \\text{ and } \\beta=n-r+1, X_{r} \\sim Beta\\alpha,\\beta
distribution relationship from: Standard Uniform 1
distribution relationship to: Beta 1
distribution relationship definition in Latex: aX_1, X_2,... , X_n (iid) \sim StandardNormal1 \Rightarrow \text{ with } \alpha=r \text{ and } \beta=n-r+1, X_{(r)} \sim Beta(\alpha,\beta)
distribution relationship pair expression Latex: StandardUniform1(0,1) \rightarrow Beta1(alpha,beta)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformBeta.pdf}

parameter has definition expressed in MathML
type: DatatypeProperty

rate of Borel-1
type: mathematical object
has name: rate of Borel-1
has code name: Borel1.shape
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in (0,1)
has short name: rate
has short code name: shape

PDF of Standard Triangular 1
type: probability density function
has name: PDF of Standard Triangular 1
has Latex expression: \begin{cases} x+1 & \text{ for } -1 < x < 0 \\ 1-x & \text{ for } 0 \leq x < 1 \end{cases}
has R code expression: x+1 for -1 < x < 0 \\ 1-x for 0 <= x < 1

mean of Standard Normal 1
type: mathematical object
has name: mean of Standard Normal 1
has Latex expression: 0

mean of Lomax 1
type: mathematical object
has name: mean of Lomax 1
has Latex expression: \frac{\lambda}{\alpha -1} \text{ for } \alpha > 1 \text { otherwise undefined }

median of Pareto Type I
type: mathematical object
has name: median of Pareto Type I
has Latex expression: x_m \sqrt[\alpha]{2}

shape parameter of Noncentral-Beta-1
type: mathematical object
has name: shape parameter of Noncentral-Beta-1
has code name: NoncentralBeta1.shape1
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: shape parameter
has short code name: shape1

Special case, Reparameterisation
type: Class
subclass of: relationship between distributions

standard deviation of Exponentially-modified-Gaussian-1
type: mathematical object
has name: standard deviation of Exponentially-modified-Gaussian-1
has code name: ExponentiallyModifiedGaussian1.stdev
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: standard deviation
has short code name: stdev

PROB k0001443
type: Transformation
label: relationship between Scaled Inverse Chi-Square and Inverse Chi-Square whereby \\mbox{If } X \\sim \\mbox{Scale-inv-}\\chi^2\\nu, \\tau^2 \\mbox{ then } \\frac{X}{\\tau^2 \\nu} \\sim \\mbox{inv-}\\chi^2\\nu
distribution relationship from: Scaled Inverse Chi-Square
distribution relationship to: Inverted-chi-square
distribution relationship definition in Latex: \mbox{If } X \sim \mbox{Scale-inv-}\chi^2(\nu, \tau^2) \mbox{ then } \frac{X}{\tau^2 \nu} \sim \mbox{inv-}\chi^2(\nu)
distribution relationship pair expression Latex: ScaledInverseChiSquare1 \rightarrow InverseChiSquare1
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution}

location of Skew-Normal
type: mathematical object
has name: location of Skew-Normal
has code name: SkewNormal1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

PDF of Half-normal 1
type: probability density function
has name: PDF of Half-normal 1
has Latex expression: \frac{2\theta}{\pi} e^{-\theta^2 x^2 / \pi}
has R code expression: 2*theta/pi * exp(-theta^2 * x^2 / pi)

shape of Generalized-Gamma-1
type: mathematical object
has name: shape of Generalized-Gamma-1
has code name: GeneralizedGamma1.shape1
has Latex symbol: d
parameter has definition expressed in plain text: d > 0
has short name: shape
has short code name: shape1

scale of Gumbel-1
type: mathematical object
has name: scale of Gumbel-1
has code name: Gumbel1.scale
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta>0, \beta \in R
has short name: scale
has short code name: scale

PROB k0000928
type: Transformation
label: relationship between Normal 1 and Chi-squared 1 whereby \\text{If } X_i \\sim N\\mu,\\sigma, i=1,2,...,n \\text{ are mutually independent and identically}\\\\\\text{ distributed random variables and } Y=\\sum_{i=1}^n X_i - \\mu/\\sigma^2 \\Rightarrow Y \\sim ChiSquared1n
distribution relationship from: Normal
distribution relationship to: Chi-square
distribution relationship definition in Latex: \text{If } X_i \sim N(\mu,\sigma), i=1,2,...,n \text{ are mutually independent and identically}\\\text{ distributed random variables and } Y=\sum_{i=1}^n ((X_i - \mu)/\sigma)^2 \Rightarrow Y \sim ChiSquared1(n)
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow ChiSquared1(n)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalChisquare.pdf}

PDF of Cauchy 1
type: probability density function
has name: PDF of Cauchy 1
has Latex expression: \frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}
has R code expression: 1 / (pi*gamma*(1 + ((x-x0)^2/gamma^2)))

mean of Gamma 2
type: mathematical object
has name: mean of Gamma 2
has Latex expression: r / \mu

relationship between Bernoulli 1 and Bernoulli 2 whereby \\alpha = \\logp/1-p
type: Reparameterisation
distribution relationship from: Bernoulli 1
distribution relationship to: Bernoulli 2
distribution relationship definition in Latex: \alpha = \log(p/(1-p))
distribution relationship pair expression Latex: Bernoulli1(p) \rightarrow Bernoulli2(\alpha)
distribution relationship reference: ProbOnto spec

has plain text expression
type: DatatypeProperty

Half Cauchy 1
type: probability distribution
has name: Half Cauchy 1
has code name: HalfCauchy1
distribution has PDF: PDF of Half Cauchy 1
distribution has CDF: CDF of Half Cauchy 1
distribution has SF: SF of Half Cauchy 1
distribution has mean: mean of Half Cauchy 1
distribution has median: median of Half Cauchy 1
distribution has variance: variance of Half Cauchy 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)

category probabilities of Categorical-Nonordered-1
type: mathematical object
has name: category probabilities of Categorical-Nonordered-1
has code name: CategoricalNonordered1.categoryProb
has Latex symbol: p_1, \ldots, p_k
parameter has definition expressed in plain text: 0 \leq p_i \leq 1, \Sigma p_i = 1
has short name: category probabilities
has short code name: categoryProb

mean of Log-Normal 2
type: mathematical object
has name: mean of Log-Normal 2
has Latex expression: e^{\mu+v/2}

mode of Standard Uniform 1
type: mathematical object
has name: mode of Standard Uniform 1
has Latex expression: \text{any value in }[0,1]

shape parameter of Johnson-SL-1
type: mathematical object
has name: shape parameter of Johnson-SL-1
has code name: JohnsonSL1.shape2
has Latex symbol: \delta
parameter has definition expressed in plain text: \delta > 0
has short name: shape parameter
has short code name: shape2

LKJ Correlation 1
type: probability distribution
has name: LKJ Correlation 1
has code name: LKJCorrelation1
distribution has PDF: PDF of LKJ Correlation 1
has variate symbol in Latex: \Sigma
has Latex expression of support: \text{positive-definite, symmetric matrix with unit diagonal (i.e., a correlation matrix)}
distribution has parameter: shape of LKJ-Correlation-1

median / geometric mean of Log-Normal-6
type: mathematical object
has name: median / geometric mean of Log-Normal-6
has code name: LogNormal6.median
has Latex symbol: m
parameter has definition expressed in plain text: m>0
has short name: median / geometric mean
has short code name: median

CDF of Generalized Poisson 1
type: cumulative density function
has name: CDF of Generalized Poisson 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

CDF of Logistic 1
type: cumulative density function
has name: CDF of Logistic 1
has Latex expression: \frac{1}{1+e^{-\frac{x-\mu}{s}}}
has R code expression: 1/(1+exp(-(x-mu)/s))

noncentrality parameter of Noncentral-Beta-1
type: mathematical object
has name: noncentrality parameter of Noncentral-Beta-1
has code name: NoncentralBeta1.noncentrality
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \geq 0
has short name: noncentrality parameter
has short code name: noncentrality

Half-normal 2
type: probability distribution
has name: Half-normal 2
has code name: HalfNormal2
distribution has PDF: PDF of Half-normal 2
distribution has CDF: CDF of Half-normal 2
distribution has mean: mean of Half-normal 2
distribution has variance: variance of Half-normal 2
has variate symbol in Latex: x
has Latex expression of support: x \in [\mu,+\infty)
distribution has parameter: scale of Half-normal-2
distribution has parameter: location of Half-normal-2

relationship between Amoroso 1 and Gamma 1 whereby \\text{Gamma1 distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when} a = 0, \\beta=1
type: Special case
distribution relationship from: Amoroso 1
distribution relationship to: Gamma 1
distribution relationship definition in Latex: \text{Gamma1 distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when} a = 0, \beta=1
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Gamma1(k,\theta)
distribution relationship reference: \cite{crooks2010amoroso}

Half-normal 1
type: probability distribution
has name: Half-normal 1
has code name: HalfNormal1
distribution has PDF: PDF of Half-normal 1
distribution has CDF: CDF of Half-normal 1
distribution has mean: mean of Half-normal 1
distribution has variance: variance of Half-normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: inverse scale of Half-normal-1

Reparameterisation, Limiting
type: Class
subclass of: relationship between distributions

relationship between Negative Binomial 3 and Negative Binomial 2 whereby \\lambda=\\mu, \\tau=1/\\phi
type: Reparameterisation
distribution relationship from: Negative Binomial 3
distribution relationship to: Negative Binomial 2
distribution relationship definition in Latex: \lambda=\mu, \tau=1/\phi
distribution relationship pair expression Latex: NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial2(\lambda, \tau)
distribution relationship reference: ProbOnto spec

variance of Gamma 2
type: mathematical object
has name: variance of Gamma 2
has Latex expression: r / \mu^2

Nakagami 1
type: probability distribution
has name: Nakagami 1
has code name: Nakagami1
distribution has PDF: PDF of Nakagami 1
distribution has CDF: CDF of Nakagami 1
distribution has mean: mean of Nakagami 1
distribution has median: median of Nakagami 1
distribution has mode: mode of Nakagami 1
distribution has variance: variance of Nakagami 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: spread of Nakagami-1
distribution has parameter: shape of Nakagami-1

shape of Generalized-Gamma-1
type: mathematical object
has name: shape of Generalized-Gamma-1
has code name: GeneralizedGamma1.shape2
has Latex symbol: p
parameter has definition expressed in plain text: p > 0
has short name: shape
has short code name: shape2

mean of Frechet 1
type: mathematical object
has name: mean of Frechet 1
has Latex expression: \begin{cases} \ \sigma\Gamma\left(1-\frac{1}{\alpha}\right) & \text{for } \alpha>1 \\ \ \infty & \text{otherwise} \end{cases}

rate or inverse scale of Exponentially-modified-Gaussian-1
type: mathematical object
has name: rate or inverse scale of Exponentially-modified-Gaussian-1
has code name: ExponentiallyModifiedGaussian1.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0
has short name: rate or inverse scale
has short code name: rate

Breit-Wigner
type: probability distribution
label: Cauchy 1
label: Lorentz
has name: Cauchy 1
has code name: Cauchy1
distribution has PDF: PDF of Cauchy 1
distribution has CDF: CDF of Cauchy 1
distribution has mean: mean of Cauchy 1
distribution has median: median of Cauchy 1
distribution has mode: mode of Cauchy 1
distribution has variance: variance of Cauchy 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location of Cauchy-1
distribution has parameter: scale of Cauchy-1

variance of Categorical Nonordered 1
type: mathematical object
has name: variance of Categorical Nonordered 1
has Latex expression: Var([x=i]) = p_i (1-p_i); \quad Cov([x=i],[x=j]) = - p_i p_j~~(i\neq j)

has Latex expression
type: DatatypeProperty

median of Standard Uniform 1
type: mathematical object
has name: median of Standard Uniform 1
has Latex expression: 0.5

median of Log-Normal 2
type: mathematical object
has name: median of Log-Normal 2
has Latex expression: e^{\mu}

PDF of Half Cauchy 1
type: probability density function
has name: PDF of Half Cauchy 1
has Latex expression: \frac2\pi \frac1{1+x^2}
has R code expression: 2/pi * 1 / (1 + x^2)

shape parameter of Johnson-SL-1
type: mathematical object
has name: shape parameter of Johnson-SL-1
has code name: JohnsonSL1.shape1
has Latex symbol: \gamma
parameter has definition expressed in plain text: \gamma \in R
has short name: shape parameter
has short code name: shape1

lognormal
type: probability distribution
label: Log-Normal 4
label: Galton
has name: Log-Normal 4
has code name: LogNormal4
distribution has PDF: PDF of Log-Normal 4
distribution has CDF: CDF of Log-Normal 4
distribution has mean: mean of Log-Normal 4
distribution has median: median of Log-Normal 4
distribution has mode: mode of Log-Normal 4
distribution has variance: variance of Log-Normal 4
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: coefficient of variation of Log-Normal-4
distribution has parameter: median / geometric mean of Log-Normal-4

PDF of LKJ Correlation 1
type: probability density function
has name: PDF of LKJ Correlation 1
has Latex expression: \propto \det(\Sigma)^{(\eta-1)}

PMF of Generalized Poisson 1
type: probability mass function
has name: PMF of Generalized Poisson 1
has Latex expression: \frac{\theta (\theta+\delta k)^{k-1}\times e^{-\theta - \delta k}}{k!}
has R code expression: (theta*(theta+k*delta)^(k-1) * exp(-theta-k*delta)) / factorial(k)

shape of Beta-1
type: mathematical object
has name: shape of Beta-1
has code name: Beta1.beta
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape
has short code name: beta

distribution relationship reference
type: DatatypeProperty

shape of Skew-Normal
type: mathematical object
has name: shape of Skew-Normal
has code name: SkewNormal1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha \in R
has short name: shape
has short code name: shape

median of Frechet 1
type: mathematical object
has name: median of Frechet 1
has Latex expression: \frac{\sigma}{\sqrt[\alpha]{\log_e(2)}}

variance of Exponentially modified Gaussian 1
type: mathematical object
has name: variance of Exponentially modified Gaussian 1
has Latex expression: \sigma^2 + 1/\lambda^2

CDF of Noncentral Beta 1
type: cumulative density function
has name: CDF of Noncentral Beta 1
has Latex expression: \sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!} I_x \left(\alpha + j,\beta\right)
has R code expression: CDFsum = function(x,alpha,beta,lambda,fromValue,toValue) { CDF=array(0,length(x)); for(j in 1:length(x)) { for(k in fromValue:toValue) { CDF[j] = CDF[j] + exp(-lambda/2) * (lambda/2)^k / factorial(k) * Rbeta(x[j],alpha+k,beta,lower=T) } } return(CDF) }

variance of Generalized Poisson 1
type: mathematical object
has name: variance of Generalized Poisson 1
has Latex expression: \frac{\theta}{(1 -\delta)^3}

variance of Normal-2
type: mathematical object
has name: variance of Normal-2
has code name: Normal2.var
has Latex symbol: v
parameter has definition expressed in plain text: v>0
has short name: variance
has short code name: var

shape of Gamma-2
type: mathematical object
has name: shape of Gamma-2
has code name: Gamma2.shape
has Latex symbol: r
parameter has definition expressed in plain text: r > 0
has short name: shape
has short code name: shape

relationship between Generalized Gamma 2 and Generalized Gamma 1 whereby a = 0, kc=d, \\text{ and rename } k=p, b=a
type: Reparameterisation
distribution relationship from: Generalized Gamma 2
distribution relationship to: Generalized Gamma 1
distribution relationship definition in Latex: a = 0, kc=d, \text{ and rename } k=p, b=a
distribution relationship pair expression Latex: GeneralizedGamma2(a,b,c,k) \rightarrow GeneralizedGamma1(a,d,p)
distribution relationship reference: ProbOnto spec

distribution has PDF
type: ObjectProperty

Poisson intensity of Conway-Maxwell-Poisson-1
type: mathematical object
has name: Poisson intensity of Conway-Maxwell-Poisson-1
has code name: ConwayMaxwellPoisson1.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda \in R, \lambda > 0
has short name: Poisson intensity
has short code name: rate

relationship between Chi 1 and Chi-squared 1 whereby \\text{If }X \\sim Chi1k \\text{ then } X^2 \\sim ChiSquared1k
type: Transformation
distribution relationship from: Chi 1
distribution relationship to: Chi-square
distribution relationship definition in Latex: \text{If }X \sim Chi1(k) \text{ then } X^2 \sim ChiSquared1(k)
distribution relationship pair expression Latex: Chi1(k) \rightarrow ChiSquared1(k)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Chi_distribution}

CDF of Half Cauchy 1
type: cumulative density function
has name: CDF of Half Cauchy 1
has Latex expression: \frac2\pi \arctan(x)
has R code expression: 2/pi * atan(x)

mode of Dirichlet 1
type: mathematical object
has name: mode of Dirichlet 1
has Latex expression: x_i = \frac{\alpha_i - 1}{\sum_{i=1}^K\alpha_i - K}, \quad \alpha_i > 1

mean of Standard Uniform 1
type: mathematical object
has name: mean of Standard Uniform 1
has Latex expression: 0.5

mode of Categorical Nonordered 1
type: mathematical object
has name: mode of Categorical Nonordered 1
has Latex expression: i\text{ such that }p_i=\max(p_1, \ldots, p_k)

Standard Normal 1
type: probability distribution
label: Standard Gaussian
has name: Standard Normal 1
has code name: StandardNormal1
distribution has PDF: PDF of Standard Normal 1
distribution has CDF: CDF of Standard Normal 1
distribution has SF: SF of Standard Normal 1
distribution has HF: HF of Standard Normal 1
distribution has mean: mean of Standard Normal 1
distribution has median: median of Standard Normal 1
distribution has mode: mode of Standard Normal 1
distribution has variance: variance of Standard Normal 1
has variate symbol in Latex: x
has Latex expression of support: x \in R
distribution has parameter: mean of Standard-Normal-1
distribution has parameter: standard deviation of Standard-Normal-1

shape of LKJ-Correlation-1
type: mathematical object
has name: shape of LKJ-Correlation-1
has code name: LKJCorrelation1.shape
has Latex symbol: \eta
parameter has definition expressed in plain text: \eta > 0
has short name: shape
has short code name: shape

PDF of Log-Normal 4
type: probability density function
has name: PDF of Log-Normal 4
has Latex expression: \frac{1}{x\sqrt{\log(cv^2+1)}\sqrt{2\pi}}\ e^{-\frac{\left[\log (x/m)\right]^2}{2\ln(cv^2+1)}}
has R code expression: 1/(x*sqrt(log(cv^2+1))*sqrt(2*pi)) * exp( -(log(x/m))^2 / (2*log(cv^2+1)) )

PDF of Log-Normal 2
type: probability density function
has name: PDF of Log-Normal 2
has Latex expression: \frac{1}{x\sqrt{v}\sqrt{2\pi}}\ e^{-\frac{\left(\log x-\mu\right)^2}{2 v}}
has R code expression: 1/(x*sqrt(v)*sqrt(2*pi)) * exp(-(ln(x)-mu)^2/(2*v))

distribution relationship pair expression Latex
type: DatatypeProperty

location of Pareto-Type-II
type: mathematical object
has name: location of Pareto-Type-II
has code name: ParetoTypeII1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

scale of Skew-Normal
type: mathematical object
has name: scale of Skew-Normal
has code name: SkewNormal1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: scale
has short code name: scale

PROB k0001444
type: Special case
label: relationship between Amoroso 1 and Levy 1 whereby \\text{Levy1}\\mu,c \\text{ distribution is a special case of the Amoroso1} a,\\theta,\\alpha,\\beta \\text{ distribution when } \\theta=c/2, \\alpha=1/2, \\beta=-1
distribution relationship from: Amoroso 1
distribution relationship to: Levy 1
distribution relationship definition in Latex: \text{Levy1}(\mu,c) \text{ distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when } \theta=c/2, \alpha=1/2, \beta=-1
distribution relationship pair expression Latex: Amoroso1(a,\theta,\alpha,\beta) \rightarrow Levy1(\mu,c)
distribution relationship reference: \cite{crooks2010amoroso}

shape of Beta-1
type: mathematical object
has name: shape of Beta-1
has code name: Beta1.alpha
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0
has short name: shape
has short code name: alpha

PDF of Noncentral Beta 1
type: probability density function
has name: PDF of Noncentral Beta 1
has Latex expression: \sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!}\frac{x^{\alpha + j - 1}\left(1-x\right)^{\beta - 1}}{\mathrm{B}\left(\alpha + j,\beta\right)}
has R code expression: PDFsum = function(x,alpha,beta,lambda,fromValue,toValue) { PDF=array(0,length(x)); for(i in 1:length(x)) { for(j in fromValue:toValue) { PDF[i] = PDF[i] + exp(-lambda/2) * (lambda/2)^j / factorial(j) * (x[i]^(alpha+j-1)*(1-x[i])^(beta-1)) / beta(alpha+j,beta) } } return(PDF) }

mean of Exponentially-modified-Gaussian-1
type: mathematical object
has name: mean of Exponentially-modified-Gaussian-1
has code name: ExponentiallyModifiedGaussian1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: mean

mode of Frechet 1
type: mathematical object
has name: mode of Frechet 1
has Latex expression: \sigma\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}

relationship between Beta 1 and Normal 1 whereby \\alpha = \\beta, \\beta \\rightarrow \\infty
type: Limiting
distribution relationship from: Beta 1
distribution relationship to: Normal
distribution relationship definition in Latex: \alpha = \beta, \beta \rightarrow \infty
distribution relationship pair expression Latex: Beta1(alpha,beta) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BetaNormal.pdf}

mean of Normal-2
type: mathematical object
has name: mean of Normal-2
has code name: Normal2.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: mean

rate of Gamma-2
type: mathematical object
has name: rate of Gamma-2
has code name: Gamma2.rate
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu > 0
has short name: rate
has short code name: rate

relationship between Frechet 1 and Weibull 1 whereby X \\sim Frechet1\\alpha,\\sigma \\rightarrow X^{-1} \\sim Weibull\\lambda=1/\\sigma,k=\\alpha
type: Transformation
distribution relationship from: Frechet 1
distribution relationship to: Weibull 1
distribution relationship definition in Latex: X \sim Frechet1(\alpha,\sigma) \rightarrow X^{-1} \sim Weibull(\lambda=1/\sigma,k=\alpha)
distribution relationship pair expression Latex: Frechet1(\alpha,\sigma) \rightarrow Weibull1(\lambda,k)
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution} \\ \cite{stan-manual:2015b}

location parameter of Johnson-SL-1
type: mathematical object
has name: location parameter of Johnson-SL-1
has code name: JohnsonSL1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location parameter
has short code name: location

median of Categorical Nonordered 1
type: mathematical object
has name: median of Categorical Nonordered 1
has Latex expression: i\text{ such that }\sum_{j=1}^{i-1} p_j \leq 0.5\text{ and }\sum_{j=1}^{i} p_j \geq 0.5

CDF of Log-Normal 2
type: cumulative density function
has name: CDF of Log-Normal 2
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\mu}{\sqrt{2}\sqrt{var}}\Big]
has R code expression: 1/2 + 1/2 * erf( (log(x)-mu) / (sqrt(2)*sqrt(var)) )

SF of Standard Uniform 1
type: survival function
has name: SF of Standard Uniform 1
has Latex expression: 1-x
has R code expression: 1-x

has MathML expression
type: DatatypeProperty

relationship between Cauchy 1 and Standard Cauchy 1 whereby x0=0, \\gamma=1
type: Special case
distribution relationship from: Breit-Wigner
distribution relationship to: Breit-Wigner
distribution relationship definition in Latex: x0=0, \gamma=1
distribution relationship pair expression Latex: Cauchy1(x_0,\gamma) \rightarrow StandardCauchy1
distribution relationship reference: \url{https://en.wikipedia.org/wiki/Cauchy_distribution}

shape of Log-Normal-6
type: mathematical object
has name: shape of Log-Normal-6
has code name: LogNormal6.geomStdev
has Latex symbol: \sigma_g
parameter has definition expressed in plain text: \sigma_g > 0
has short name: shape
has short code name: geomStdev

shape of Pareto-Type-I
type: mathematical object
has name: shape of Pareto-Type-I
has code name: ParetoTypeI1.shape
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0, \alpha \in R
has short name: shape
has short code name: shape

mean of Generalized Poisson 1
type: mathematical object
has name: mean of Generalized Poisson 1
has Latex expression: \frac{\theta}{1 -\delta}

CDF of Log-Normal 4
type: cumulative density function
has name: CDF of Log-Normal 4
has Latex expression: \frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log m}{\sqrt{2}\sqrt{\log(cv^2+1)}}\Big]
has R code expression: 1/2 + 1/2 * erf( (log(x)-log(m)) / (sqrt(2*log(cv^2+1))) )

mean of Wishart 1
type: mathematical object
has name: mean of Wishart 1
has Latex expression: n V

variance of Categorical Ordered 1
type: mathematical object
has name: variance of Categorical Ordered 1
has Latex expression: Var([x=i]) = p_i (1-p_i); \quad Cov([x=i],[x=j]) = - p_i p_j~~(i\neq j)

mean of Zero-Inflated-Negative-Binomial-1
type: mathematical object
has name: mean of Zero-Inflated-Negative-Binomial-1
has code name: ZeroInflatedNegativeBinomial1.rate
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0
has short name: mean
has short code name: rate

relationship between Bernoulli 1 and Binomial 1 whereby \\Sigma X iid
type: Transformation
distribution relationship from: Bernoulli 1
distribution relationship to: Binomial 1
distribution relationship definition in Latex: \Sigma X (iid)
distribution relationship pair expression Latex: Bernoulli1(p) \rightarrow Binomial1(n,p)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BernoulliBinomial.pdf}

random variable
type: Class
subclass of: mathematical object

Johnson SL 1
type: probability distribution
has name: Johnson SL 1
has code name: JohnsonSL1
distribution has PDF: PDF of Johnson SL 1
distribution has CDF: CDF of Johnson SL 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: location parameter of Johnson-SL-1
distribution has parameter: shape parameter of Johnson-SL-1
distribution has parameter: scale parameter of Johnson-SL-1

mode of Normal 2
type: mathematical object
has name: mode of Normal 2
has Latex expression: \mu

variance of Standard Triangular 1
type: mathematical object
has name: variance of Standard Triangular 1
has Latex expression: 42375

CDF of Weibull 2
type: cumulative density function
has name: CDF of Weibull 2
has Latex expression: 1- \exp(-x^v \lambda)
has R code expression: 1- exp(-x^v * lambda)

CDF of GeneralizedPoisson2
type: cumulative density function
has name: CDF of GeneralizedPoisson2
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

Gompertz 1
type: probability distribution
has name: Gompertz 1
has code name: Gompertz1
distribution has PDF: PDF of Gompertz 1
distribution has CDF: CDF of Gompertz 1
distribution has SF: SF of Gompertz 1
distribution has HF: HF of Gompertz 1
distribution has median: median of Gompertz 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: shape of Gompertz-1
distribution has parameter: scale of Gompertz-1

shape2 of Generalized-Negative-Binomial-1
type: mathematical object
has name: shape2 of Generalized-Negative-Binomial-1
has code name: GeneralizedNegativeBinomial1.m
has Latex symbol: m
parameter has definition expressed in plain text: m > 0
has short name: shape2
has short code name: m

shape parameter of Muth-1
type: mathematical object
has name: shape parameter of Muth-1
has code name: Muth1.shape
has Latex symbol: \kappa
parameter has definition expressed in plain text: 0 \leq \kappa \leq 1
has short name: shape parameter
has short code name: shape

PDF of Frechet 2
type: probability density function
has name: PDF of Frechet 2
has Latex expression: \frac{\alpha}{\sigma} \Big(\frac{x-m}{\sigma}\Big)^{-\alpha-1} \exp\Big(-\Big(\frac{x-m}{\sigma}\Big)^{-\alpha}\Big)
has R code expression: alpha/sigma * ((x-m)/sigma)^(-alpha-1) * exp(-((x-m)/sigma)^(-alpha))

variance of Normal 2
type: mathematical object
has name: variance of Normal 2
has Latex expression: v

number of values of Uniform-Discrete-2
type: mathematical object
has name: number of values of Uniform-Discrete-2
has code name: UniformDiscrete2.numberOfValues
has Latex symbol: n
parameter has definition expressed in plain text: n \in N
has short name: number of values
has short code name: numberOfValues

distribution has CDF
type: ObjectProperty

mean of Nakagami 1
type: mathematical object
has name: mean of Nakagami 1
has Latex expression: \frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)} \Big(\frac{\Omega}{m}\Big)^{\frac12}

Exponential 1
type: probability distribution
label: Negative exponential
has name: Exponential 1
has code name: Exponential1
distribution has PDF: PDF of Exponential 1
distribution has CDF: CDF of Exponential 1
distribution has SF: SF of Exponential 1
distribution has HF: HF of Exponential 1
distribution has mean: mean of Exponential 1
distribution has median: median of Exponential 1
distribution has mode: mode of Exponential 1
distribution has variance: variance of Exponential 1
has variate symbol in Latex: x
has Latex expression of support: x \in [0,+\infty)
distribution has parameter: rate or inverse scale of Exponential-1

median of Maxwell Boltzmann 1
type: mathematical object
has name: median of Maxwell Boltzmann 1
has Latex expression: \sqrt{2} a

variance of Beta 1
type: mathematical object
has name: variance of Beta 1
has Latex expression: \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}

dispersion of Generalized-Poisson-1
type: mathematical object
has name: dispersion of Generalized-Poisson-1
has code name: GeneralizedPoisson1.dispersion
has Latex symbol: \delta
parameter has definition expressed in plain text: max(-1,-\theta/m) < \delta < 1 \text{ with } m (\geq 4) \text { the largest positive integer for which } \theta + m \delta > 0 \text{ when } \delta < 0
has short name: dispersion
has short code name: dispersion

scale parameter of Johnson-SU-1
type: mathematical object
has name: scale parameter of Johnson-SU-1
has code name: JohnsonSU1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma > 0
has short name: scale parameter
has short code name: scale

relationship between Normal 2 and Normal 3 whereby \\mu = \\mu, \\tau = 1 / v
type: Reparameterisation
distribution relationship from: Normal
distribution relationship to: Gaussian
distribution relationship definition in Latex: \mu = \mu, \tau = 1 / v
distribution relationship pair expression Latex: Normal2(\mu,v) \rightarrow Normal3(\mu,\tau)
distribution relationship reference: ProbOnto spec

category probabilities of Categorical-Ordered-1
type: mathematical object
has name: category probabilities of Categorical-Ordered-1
has code name: CategoricalOrdered1.categoryProb
has Latex symbol: p_1, \ldots, p_k
parameter has definition expressed in plain text: 0 \leq p_i \leq 1, \Sigma p_i = 1
has short name: category probabilities
has short code name: categoryProb

PDF of Exponential 1
type: probability density function
has name: PDF of Exponential 1
has Latex expression: \lambda e^{-\lambda x}
has R code expression: lambda*exp(-lambda*x)

variance of Zipf 1
type: mathematical object
has name: variance of Zipf 1
has Latex expression: (H_{n,\alpha-2} H_{n,\alpha} - H_{n,\alpha-1}^2)/ H_{n,\alpha}^2

mean of Multinomial 1
type: mathematical object
has name: mean of Multinomial 1
has Latex expression: E\{X_i\} = np_i

relationship between Normal 1 and Normal 2 whereby \\mu = \\mu, v = \\sigma^2
type: Reparameterisation
distribution relationship from: Normal
distribution relationship to: Normal
distribution relationship definition in Latex: \mu = \mu, v = \sigma^2
distribution relationship pair expression Latex: Normal1(\mu,\sigma) \rightarrow Normal2(\mu,v)
distribution relationship reference: ProbOnto spec

variance of Epanechnikov 1
type: mathematical object
has name: variance of Epanechnikov 1
has Latex expression: \frac15

mean of Zero-Inflated Negative Binomial 1
type: mathematical object
has name: mean of Zero-Inflated Negative Binomial 1
has Latex expression: \mu

variate
type: Class
subclass of: mathematical object

mean of Dirichlet 1
type: mathematical object
has name: mean of Dirichlet 1
has Latex expression: E[X_i] = \frac{\alpha_i}{\sum_k \alpha_k}

PDF of Gompertz 1
type: probability density function
has name: PDF of Gompertz 1
has Latex expression: b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx} \right)
has R code expression: b*eta*exp(b*x)*exp(eta)*exp(-eta*exp(b*x))

shape of LKJ-Correlation-2
type: mathematical object
has name: shape of LKJ-Correlation-2
has code name: LKJCorrelation2.shape
has Latex symbol: \eta
parameter has definition expressed in plain text: \eta > 0
has short name: shape
has short code name: shape

mode of Standard Triangular 1
type: mathematical object
has name: mode of Standard Triangular 1
has Latex expression: 0

HF of Weibull 2
type: hazard function
has name: HF of Weibull 2
has Latex expression: v\lambda \,x^{v-1}
has R code expression: v*lambda * x^(v-1)

Inverse Weibull
type: probability distribution
label: Frechet 2
has name: Frechet 2
has code name: Frechet2
distribution has PDF: PDF of Frechet 2
distribution has CDF: CDF of Frechet 2
distribution has mean: mean of Frechet 2
distribution has median: median of Frechet 2
distribution has mode: mode of Frechet 2
distribution has variance: variance of Frechet 2
has variate symbol in Latex: x
has Latex expression of support: x \in (m, \infty)
distribution has parameter: shape of Frechet-2
distribution has parameter: scale of Frechet-2
distribution has parameter: location of minimum of Frechet-2

mean of GeneralizedPoisson2
type: mathematical object
has name: mean of GeneralizedPoisson2
has Latex expression: \mu

relationship between Inverse Gaussian 1 and Standard Normal 1 whereby \\lambda \\rightarrow \\infty
type: Limiting
distribution relationship from: Inverse Gaussian 1
distribution relationship to: Standard Normal 1
distribution relationship definition in Latex: \lambda \rightarrow \infty
distribution relationship pair expression Latex: InverseGaussian1(\lambda,\mu) \rightarrow StandardNormal1(0,1)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/InversegaussianStandardnormal.pdf}

Generalized Poisson 3
type: probability distribution
has name: Generalized Poisson 3
has code name: GeneralizedPoisson3
distribution has PMF: PMF of Generalized Poisson 3
distribution has CDF: CDF of Generalized Poisson 3
distribution has mean: mean of Generalized Poisson 3
distribution has variance: variance of Generalized Poisson 3
has variate symbol in Latex: y
has Latex expression of support: y \in \{0,1,2,3,\dots\}
distribution has parameter: dispersion of Generalized-Poisson-3
distribution has parameter: mean of Generalized-Poisson-3

median of Nakagami 1
type: mathematical object
has name: median of Nakagami 1
has Latex expression: \sqrt{\Omega}

distribution has PMF
type: ObjectProperty

mean of Maxwell Boltzmann 1
type: mathematical object
has name: mean of Maxwell Boltzmann 1
has Latex expression: 2a \sqrt{\frac{2}{\pi}}

mode of Beta 1
type: mathematical object
has name: mode of Beta 1
has Latex expression: \frac{\alpha-1}{\alpha+\beta-2}

lambda of Normal-inverse-gamma-1
type: mathematical object
has name: lambda of Normal-inverse-gamma-1
has code name: NormalInverseGamma1.lambda
has Latex symbol: \lambda
parameter has definition expressed in plain text: \lambda > 0, \lambda \in R
has short name: lambda
has short code name: lambda

Poisson intensity of Generalized-Poisson-1
type: mathematical object
has name: Poisson intensity of Generalized-Poisson-1
has code name: GeneralizedPoisson1.rate
has Latex symbol: \theta
parameter has definition expressed in plain text: \theta > 0
has short name: Poisson intensity
has short code name: rate

shape1 of Generalized-Negative-Binomial-1
type: mathematical object
has name: shape1 of Generalized-Negative-Binomial-1
has code name: GeneralizedNegativeBinomial1.beta
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta = 0 \text{ or } 1 \le \beta \le \theta^{-1}
has short name: shape1
has short code name: beta

location parameter of Johnson-SU-1
type: mathematical object
has name: location parameter of Johnson-SU-1
has code name: JohnsonSU1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location parameter
has short code name: location

PDF of Wishart 1
type: probability density function
has name: PDF of Wishart 1
has Latex expression: \frac{|X|^{\frac{n-p-1}{2}} e^{-\frac{\text{tr}(V^{-1}X)}{2}}}{2^\frac{np}{2}|V|^\frac{n}{2}\Gamma_p(\frac{n}{2})}

PDF of Generalized Gamma 3
type: probability density function
has name: PDF of Generalized Gamma 3
has Latex expression: \frac{\beta}{\Gamma(r)}\mu^{\beta r}x^{\beta r -1}\exp[-(\mu x)^\beta]
has R code expression: beta / gamma(r) * mu^(beta*r) * x^(beta*r -1) * exp(-(mu*x)^beta)

variance of Multinomial 1
type: mathematical object
has name: variance of Multinomial 1
has Latex expression: Var(X_i) = n p_i (1-p_i); \quad Cov(X_i,X_j) = - n p_i p_j~~(i\neq j)

relationship between Weibull 1 and Exponential 1 whereby k=1, \\lambda_{Exponetial} = 1/\\lambda
type: Reparameterisation
distribution relationship from: Weibull 1
distribution relationship to: Exponential 1
distribution relationship definition in Latex: k=1, \lambda_{Exponetial} = 1/\lambda
distribution relationship pair expression Latex: Weibull1(\lambda,k) \rightarrow Exponential1(\lambda_{Exponetial})
distribution relationship reference: \cite{forbes2011statistical}

Dirichlet 1
type: probability distribution
label: Dirichlet
has name: Dirichlet 1
has code name: Dirichlet1
distribution has PDF: PDF of Dirichlet 1
distribution has mean: mean of Dirichlet 1
distribution has mode: mode of Dirichlet 1
distribution has variance: variance of Dirichlet 1
has variate symbol in Latex: x
has Latex expression of support: x_1, \cdots, x_K \quad\text{where}\quad x_i \in [0,1]\quad and \quad\sum_{i=1}^K x_i = 1
distribution has parameter: concentration of Dirichlet-1

mode of Log-Normal 2
type: mathematical object
has name: mode of Log-Normal 2
has Latex expression: e^{\mu-v}

non-standardized Student's t-distribution
type: probability distribution
label: t Location-Scale
label: Student's t-distribution 3
has name: Student's t-distribution 3
has code name: StudentT3
distribution has PDF: PDF of Student's t-distribution 3
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: mean of Student-s-t-distribution-3
distribution has parameter: degrees of freedom of Student-s-t-distribution-3
distribution has parameter: scale of Student-s-t-distribution-3

location of Levy-1
type: mathematical object
has name: location of Levy-1
has code name: Levy1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

Weibull 2
type: probability distribution
has name: Weibull 2
has code name: Weibull2
distribution has PDF: PDF of Weibull 2
distribution has CDF: CDF of Weibull 2
distribution has SF: SF of Weibull 2
distribution has HF: HF of Weibull 2
has variate symbol in Latex: x
has Latex expression of support: x > 0
distribution has parameter: lambda of Weibull-2
distribution has parameter: shape of Weibull-2

variance of Bernoulli 1
type: mathematical object
has name: variance of Bernoulli 1
has Latex expression: p(1-p)

CDF of Johnson SL 1
type: cumulative density function
has name: CDF of Johnson SL 1
has Latex expression: \begin{cases} \frac12 \text{erfc}\left[-\frac{\gamma + \delta \,\log\left[\frac{x-\mu}{\sigma}\right]}{\sqrt{2}}\right] & \text{for } \mu < x \le \mu + \sigma \\ \frac12\left(1+ \text{erf}\left[\frac{\gamma + \delta \,\log\left[\frac{x-\mu}{\sigma}\right]}{\sqrt{2}}\right] \right) & \text{for } x > \mu + \sigma \end{cases}
has R code expression: CDF1=1/2*erfc(-(gamma+delta*log((x-mu)/sigma))/sqrt(2)) CDF2=1/2*(1+ erf((gamma+delta*log((x-mu)/sigma))/sqrt(2)))

CDF of Zero-Inflated Negative Binomial 1
type: cumulative density function
has name: CDF of Zero-Inflated Negative Binomial 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: c(PMF1,cumsum(PMF2)+PMF1)

PROB k0001420
type: Transformation
label: relationship between Standard Uniform 1 and Standard Power 1 whereby \\text{If } X \\sim \\text{ StandardUniform1, then } Y = X^{1/\\beta} \\text{ has the StandardPower}\\beta \\text{ distribution, where } \\beta > 1
distribution relationship from: Standard Uniform 1
distribution relationship to: Standard Power 1
distribution relationship definition in Latex: \text{If } X \sim \text{ StandardUniform1, then } Y = X^{1/\beta} \text{ has the StandardPower}(\beta) \text{ distribution, where } \beta > 1
distribution relationship pair expression Latex: StandardUniform1 \rightarrow StandardPower1(\beta)
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformStandardpowerB.pdf}

Mixture Distribution 1
type: probability distribution
has name: Mixture Distribution 1
has code name: MixtureDistribution1
distribution has PMF: PDF of Mixture Distribution 1
distribution has parameter: mixing coefficients of Mixture-Distribution-1

PMF of Generalized Poisson 3
type: probability mass function
has name: PMF of Generalized Poisson 3
has Latex expression: \Big( \frac{\mu}{1+\alpha \mu}\Big)^y \frac{(1+\alpha y)^{y-1}}{y!} \exp\Big[ \frac{-\mu (1+\alpha y)}{1+\alpha \mu}\Big]
has R code expression: (mu/(1+alpha*mu))^y *(1+alpha*y)^(y-1)/factorial(y)*exp(-mu*(1+alpha*y)/(1+alpha*mu))

PDF of Nakagami 1
type: probability density function
has name: PDF of Nakagami 1
has Latex expression: \frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1} \exp(-\frac{m}{\omega}x^2)
has R code expression: 2*m^m / (gamma(m)*Omega^m)*x^(2*m-1)*exp(-m/Omega*x^2)

PDF of Standard Power 1
type: probability density function
has name: PDF of Standard Power 1
has Latex expression: \beta x^{\beta-1}
has R code expression: (beta * x^(beta-1))

variance of GeneralizedPoisson2
type: mathematical object
has name: variance of GeneralizedPoisson2
has Latex expression: \frac{\mu}{(1 -\delta)^2}

mean of Frechet 2
type: mathematical object
has name: mean of Frechet 2
has Latex expression: \begin{cases} \ m+\sigma\Gamma\left(1-\frac{1}{\alpha}\right) & \text{for } \alpha>1 \\ \ \infty & \text{otherwise} \end{cases}

PDF of Multivariate Gaussian Process Distribution 2
type: probability density function
has name: PDF of Multivariate Gaussian Process Distribution 2
has Latex expression: \prod^K_{i=1} \text{MultivariateNormal1}(x_i|0,w^{-1}_i LL^T)

mean of Benford 1
type: mathematical object
has name: mean of Benford 1
has Latex expression: \approx 3.4402

SF of Standard Normal 1
type: survival function
has name: SF of Standard Normal 1
has Latex expression: 1 - \frac12 \text{erfc}\left(-\frac{x}{\sqrt{2}}\right)
has R code expression: 1 - 1/2 *erfc(-x/sqrt(2))

Wishart 1
type: probability distribution
has name: Wishart 1
has code name: Wishart1
distribution has PDF: PDF of Wishart 1
distribution has mean: mean of Wishart 1
distribution has mode: mode of Wishart 1
distribution has variance: variance of Wishart 1
has variate symbol in Latex: X
has Latex expression of support: X(p \times p) - \text{positive definite matrix}
distribution has parameter: scale matrix of Wishart-1
distribution has parameter: degrees of freedom of Wishart-1

PROB k0000870
type: Limiting
label: relationship between Binomial 1 and Normal 1 whereby \\text{For } X \\sim Binomial1n,p \\text{ as } n \\rightarrow \\infty, X \\text{ is approximately normally distributed } \\\\Normal1\\mu,\\sigma \\text{ with } \\mu=np, \\sigma=np1-p.
distribution relationship from: Binomial 1
distribution relationship to: Normal
distribution relationship definition in Latex: \text{For } X \sim Binomial1(n,p) \text{ as } n \rightarrow \infty, X \text{ is approximately normally distributed } \\Normal1(\mu,\sigma) \text{ with } \mu=np, \sigma=np(1-p).
distribution relationship pair expression Latex: Binomial1(n,p) \rightarrow Normal1(\mu,\sigma)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialNormal.pdf}

variance of Folded Normal 1
type: mathematical object
has name: variance of Folded Normal 1
has Latex expression: \mu^2 + \sigma^2 - \Big(\sigma \sqrt{\tfrac{2}{\pi}} \, e^{(-\mu^2/2\sigma^2)} + \mu \left(1 - 2\,\Phi(\tfrac{-\mu}{\sigma}) \right)\Big)^2

degree of freedom of F-1
type: mathematical object
has name: degree of freedom of F-1
has code name: F1.denominator
has Latex symbol: n_2
parameter has definition expressed in plain text: n_2 > 0
has short name: degree of freedom
has short code name: denominator

CDF of Maxwell Boltzmann 1
type: cumulative density function
has name: CDF of Maxwell Boltzmann 1
has Latex expression: \text{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \frac{x e^{-x^2/\left(2a^2\right)}}{a}
has R code expression: erf(x/(sqrt(2)*a)) -sqrt(2/pi) * (x * exp(-x^2/(2*a^2)) )/a

median of Beta 1
type: mathematical object
has name: median of Beta 1
has Latex expression: I_{\frac{1}{2}}^{[-1]}(\alpha,\beta)

number of trials of Multinomial-1
type: mathematical object
has name: number of trials of Multinomial-1
has code name: Multinomial1.numberOfTrials
has Latex symbol: n
parameter has definition expressed in plain text: n > 0, n \in N
has short name: number of trials
has short code name: numberOfTrials

SF of Beta 1
type: survival function
has name: SF of Beta 1
has Latex expression: 1-I_x(\alpha,\beta)
has R code expression: 1 - Rbeta(x, alpha, beta)

Standard Two-Sided Power 1
type: probability distribution
label: STSP
has name: Standard Two-Sided Power 1
has code name: StandardTwoSidedPower1
distribution has PDF: PDF of Standard Two-Sided Power 1
distribution has CDF: CDF of Standard Two-Sided Power 1
distribution has mean: mean of Standard Two-Sided Power 1
distribution has variance: variance of Standard Two-Sided Power 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,1)
distribution has parameter: location of Standard-Two-Sided-Power-1
distribution has parameter: shape of Standard-Two-Sided-Power-1

Generalized Gamma 3
type: probability distribution
has name: Generalized Gamma 3
has code name: GeneralizedGamma3
distribution has PDF: PDF of Generalized Gamma 3
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape of Generalized-Gamma-3
distribution has parameter: scale of Generalized-Gamma-3

scale of Levy-1
type: mathematical object
has name: scale of Levy-1
has code name: Levy1.scale
has Latex symbol: c
parameter has definition expressed in plain text: c > 0, c \in R
has short name: scale
has short code name: scale

PDF of Weibull 2
type: probability density function
has name: PDF of Weibull 2
has Latex expression: v\lambda \,x^{v-1} e^{-\lambda x^{v}}
has R code expression: v*lambda * x^(v-1) * exp(-lambda * x^v)

PDF of Dirichlet 1
type: probability density function
has name: PDF of Dirichlet 1
has Latex expression: \frac{1}{B(\alpha)} \prod_{i=1}^K x_i^{\alpha_i - 1} \\ \text{ where} \quad B(\alpha) = \frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K \alpha_i\bigr)} \\ \text{ where} \quad \alpha=(\alpha_1,\ldots,\alpha_K)

variance of Log-Normal 2
type: mathematical object
has name: variance of Log-Normal 2
has Latex expression: (e^v-1) e^{2\mu+v}

PMF of Zero-Inflated Negative Binomial 1
type: probability mass function
has name: PMF of Zero-Inflated Negative Binomial 1
has Latex expression: \begin{cases} p0 + (1-p0) \Big(\frac{1}{1 + \tau\lambda} \Big)^{1/\tau} & \text{for } y = 0 \\ (1-p0) \frac{\Gamma(y+1/\tau)}{y!\Gamma(1/\tau)} \Big(\frac{1}{1 + \tau\lambda} \Big)^{1/\tau} \Big(\frac{\lambda}{1/\tau + \lambda} \Big)^{y} & \text{for } y > 0 \end{cases}
has R code expression: PMF1=p0 + (1-p0) * (1/ (1 + tau * lambda))^(1/tau) # for y=0 PMF2=(1-p0) * gamma(y+1/tau) / (factorial(y) *gamma(1/tau)) * (1/(1 + tau*lambda))^(1/tau) * (lambda/(1/tau + lambda))^y # for y>0

relationship between Negative Binomial 1 and Negative Binomial 5 whereby \\alpha = r, \\beta = p / 1-p
type: Reparameterisation
distribution relationship from: Negative Binomial 1
distribution relationship to: Negative Binomial 5
distribution relationship definition in Latex: \alpha = r, \beta = p / (1-p)
distribution relationship pair expression Latex: NegativeBinomial1(r,p) \rightarrow NegativeBinomial5(\alpha, \beta)
distribution relationship reference: ProbOnto spec

probability of success of Bernoulli-1
type: mathematical object
has name: probability of success of Bernoulli-1
has code name: Bernoulli1.probability
has Latex symbol: p
parameter has definition expressed in plain text: 0<p<1, p \in R
has short name: probability of success
has short code name: probability

PDF of Johnson SL 1
type: probability density function
has name: PDF of Johnson SL 1
has Latex expression: \frac{e^{-\frac12 \left( \gamma + \delta \log\left[\frac{x-\mu}{\sigma}\right]\right)^2} \delta}{\sqrt{2\pi} (x-\mu)}\\
has R code expression: exp(-1/2*(gamma+delta*log((x-mu)/sigma))^2)*delta/(sqrt(2*pi)*(x-mu))

PDF of Student's t-distribution 3
type: probability density function
has name: PDF of Student's t-distribution 3
has Latex expression: \frac{\Gamma \left(\frac{\nu+1}{2} \right)}{\Gamma \left(\frac{\nu}{2} \right)} \frac{1}{\sqrt{\nu\pi} \sigma} \left(1+\frac{1}{\nu} \left(\frac{x-\mu}{\sigma}\right)^2 \right)^{-\frac{\nu+1}{2}}
has R code expression: gamma((nu+1)/2)/gamma(nu/2)*1/sqrt(nu*pi)/sigma*(1+1/nu*((x-mu)/sigma)^2)^(-(nu+1)/2)

relationship between Standard Uniform 1 and Benford 1 whereby \\text{If } X \\sim \\text{StandardUniform1, then } Y = \\lfloor 10^X \\rfloor \\text{ has the Benford1 distribution}
type: Transformation
distribution relationship from: Standard Uniform 1
distribution relationship to: Benford 1
distribution relationship definition in Latex: \text{If } X \sim \text{StandardUniform1, then } Y = \lfloor 10^X \rfloor \text{ has the Benford1 distribution}
distribution relationship pair expression Latex: StandardUniform1 \rightarrow Benford1
distribution relationship reference: \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformBenford.pdf}

PDF of Maxwell Boltzmann 1
type: probability density function
has name: PDF of Maxwell Boltzmann 1
has Latex expression: \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/\left(2a^2\right)}}{a^3}
has R code expression: sqrt(2/pi) * (x^2 * exp(-x^2/(2*a^2)) )/a^3

HF of Standard Normal 1
type: hazard function
has name: HF of Standard Normal 1
has Latex expression: \frac{\sqrt{\frac2\pi}}{e^{\frac{x^2}{2}} \left(2-\text{erfc}\left(\frac{-x}{\sqrt{2}}\right)\right)}
has R code expression: sqrt(2/pi)/(exp(x^2/2)*(2-erfc(-x/sqrt(2))))

degree of freedom of F-1
type: mathematical object
has name: degree of freedom of F-1
has code name: F1.numerator
has Latex symbol: n_1
parameter has definition expressed in plain text: n_1 > 0
has short name: degree of freedom
has short code name: numerator

CDF of Frechet 2
type: cumulative density function
has name: CDF of Frechet 2
has Latex expression: \exp\Big(-\Big(\frac{\sigma}{x-m}\Big)^\alpha\Big)
has R code expression: exp(-(sigma/(x-m))^alpha)

Cholesky parameterization
type: probability distribution
label: Multivariate Gaussian Process Distribution
label: Multivariate Gaussian Process Distribution 2
has name: Multivariate Gaussian Process Distribution 2
has code name: MultivariateGaussianProcess2
distribution has PDF: PDF of Multivariate Gaussian Process Distribution 2
has variate symbol in Latex: x
has Latex expression of support: x \in R^{K\times N}
distribution has parameter: lower triangular matrix of Multivariate-Gaussian-Process-Distribution-2
distribution has parameter: vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-2

CDF of Nakagami 1
type: cumulative density function
has name: CDF of Nakagami 1
has Latex expression: \frac{\gamma(m,\frac{m}{\Omega}x^2)}{\Gamma(m)}
has R code expression: Igamma(m,m/Omega*x^2,lower=T)/gamma(m)

Standard Power 1
type: probability distribution
has name: Standard Power 1
has code name: StandardPower1
distribution has PDF: PDF of Standard Power 1
distribution has CDF: CDF of Standard Power 1
distribution has SF: SF of Standard Power 1
distribution has HF: HF of Standard Power 1
distribution has mean: mean of Standard Power 1
distribution has median: median of Standard Power 1
distribution has variance: variance of Standard Power 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,+\infty)
distribution has parameter: shape parameter of Standard-Power-1

success probability of Negative-Binomial-1
type: mathematical object
has name: success probability of Negative-Binomial-1
has code name: NegativeBinomial1.probability
has Latex symbol: p
parameter has definition expressed in plain text: p \in (0,1)
has short name: success probability
has short code name: probability

relationship between Negative Binomial 1 and Negative Binomial 4 whereby p = 1 -p
type: Reparameterisation
distribution relationship from: Negative Binomial 1
distribution relationship to: Negative Binomial 4
distribution relationship definition in Latex: p = 1 -p
distribution relationship pair expression Latex: NegativeBinomial1(r,p) \rightarrow NegativeBinomial4(r,p)
distribution relationship reference: ProbOnto spec

location of Folded-Normal-1
type: mathematical object
has name: location of Folded-Normal-1
has code name: FoldedNormal1.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: location
has short code name: location

mean of Beta 1
type: mathematical object
has name: mean of Beta 1
has Latex expression: \frac{\alpha}{\alpha+\beta}

Multivariate Normal 1
type: probability distribution
label: Multivariate Gaussian 1
has name: Multivariate Normal 1
has code name: MultivariateNormal1
distribution has PDF: PDF of Multivariate Normal 1
distribution has CDF: CDF of Multivariate Normal 1
distribution has mean: mean of Multivariate Normal 1
distribution has mode: mode of Multivariate Normal 1
distribution has variance: variance of Multivariate Normal 1
has variate symbol in Latex: x
has Latex expression of support: -\infty < x_i < \infty, i = 1,...,k
distribution has parameter: covariance matrix of Multivariate-Normal-1
distribution has parameter: location of Multivariate-Normal-1

PDF of Mixture Distribution 1
type: probability density function
has name: PDF of Mixture Distribution 1
has Latex expression: f(x; \pi, \theta) = \sum_{i=1}^{K} \pi_{i}\; p_i(x; \theta_i) \text{ where } p_i(x; \theta_i) \text{ the PDF of the } i^{th} \text{ component with parameters } \theta_i

precision matrix of Multivariate-Student-T-2
type: mathematical object
has name: precision matrix of Multivariate-Student-T-2
has code name: MultivariateStudentT2.precisionMatrix
has Latex symbol: T
parameter has definition expressed in plain text: \text{Inverse of the covariance matrix}
has short name: precision matrix
has short code name: precisionMatrix

SF of Standard Triangular 1
type: survival function
has name: SF of Standard Triangular 1
has Latex expression: \begin{cases} -\frac12 x^2-x+\frac12 & \text{ for } -1 < x < 0 \\ \frac12 x^2-x+\frac12 & \text{ for } 0 \leq x < 1 \end{cases}
has R code expression: SF1=-1/2*x^2-x+1/2 SF2=1/2*x^2-x+1/2

degrees of freedom of Student-s-t-distribution-3
type: mathematical object
has name: degrees of freedom of Student-s-t-distribution-3
has code name: StudentT3.degreesOfFreedom
has Latex symbol: \nu
parameter has definition expressed in plain text: \nu > 0, \nu \in R
has short name: degrees of freedom
has short code name: degreesOfFreedom

median of Lomax 1
type: mathematical object
has name: median of Lomax 1
has Latex expression: \lambda (\sqrt[\alpha]{2} - 1)

variance of Standard Normal 1
type: mathematical object
has name: variance of Standard Normal 1
has Latex expression: 1

relationship between Log-Normal 1 and Log-Normal 3 whereby m = \\exp\\mu, \\sigma = \\sigma
type: Reparameterisation
distribution relationship from: Log-Normal 1
distribution relationship to: Log-Normal 3
distribution relationship definition in Latex: m = \exp(\mu), \sigma = \sigma
distribution relationship pair expression Latex: LogNormal1(\mu,\sigma) \rightarrow LogNormal3(m,\sigma)
distribution relationship reference: ProbOnto spec

median of Bernoulli 1
type: mathematical object
has name: median of Bernoulli 1
has Latex expression: \begin{cases} 0 & \text{if } q > p\\ 0.5 & \text{if } q=p\\ 1 & \text{if } q<p \end{cases}

minimum of Uniform-Discrete-1
type: mathematical object
has name: minimum of Uniform-Discrete-1
has code name: UniformDiscrete1.minimum
has Latex symbol: a
parameter has definition expressed in plain text: a \in \{\dots,-2,-1,0,1,2,3,\dots\}
has short name: minimum
has short code name: minimum

scalar
type: Class
subclass of: mathematical object

Zero-Inflated Negative Binomial 1
type: probability distribution
has name: Zero-Inflated Negative Binomial 1
has code name: ZeroInflatedNegativeBinomial1
distribution has PMF: PMF of Zero-Inflated Negative Binomial 1
distribution has CDF: CDF of Zero-Inflated Negative Binomial 1
distribution has mean: mean of Zero-Inflated Negative Binomial 1
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: overdispersion of Zero-Inflated-Negative-Binomial-1
distribution has parameter: probability of zero of Zero-Inflated-Negative-Binomial-1
distribution has parameter: mean of Zero-Inflated-Negative-Binomial-1

shape1 of Generalized-Gamma-2
type: mathematical object
has name: shape1 of Generalized-Gamma-2
has code name: GeneralizedGamma2.shape1
has Latex symbol: c
parameter has definition expressed in plain text: c > 0
has short name: shape1
has short code name: shape1

ordered cutpoints of Ordered-Logistic-1
type: mathematical object
has name: ordered cutpoints of Ordered-Logistic-1
has code name: OrderedLogistic1.cutpoint
has Latex symbol: c
parameter has definition expressed in plain text: c \in R^{K-1} \mbox{ such that } c_k < c_{k-1} \text{ for } k \in \{1,...,K-2\}, K \in N, K>2
has short name: ordered cutpoints
has short code name: cutpoint

Levy 1
type: probability distribution
has name: Levy 1
has code name: Levy1
distribution has PDF: PDF of Levy 1
distribution has CDF: CDF of Levy 1
distribution has mean: mean of Levy 1
distribution has median: median of Levy 1
distribution has mode: mode of Levy 1
distribution has variance: variance of Levy 1
has variate symbol in Latex: x
has Latex expression of support: x \in [\mu,+\infty)
distribution has parameter: scale of Levy-1
distribution has parameter: location of Levy-1

HF of Beta 1
type: hazard function
has name: HF of Beta 1
has Latex expression: \frac{x^{\alpha-1}(1-x)^{\beta-1}} {(1-I_x(\alpha,\beta)) B(\alpha,\beta)}
has R code expression: (x^(alpha-1) * (1-x)^(beta-1))/((1-Rbeta(x,alpha,beta))*beta(alpha,beta))

scale of Generalized-Gamma-3
type: mathematical object
has name: scale of Generalized-Gamma-3
has code name: GeneralizedGamma3.scale
has Latex symbol: r
parameter has definition expressed in plain text: r > 0
has short name: scale
has short code name: scale

degrees of freedom of Wishart-1
type: mathematical object
has name: degrees of freedom of Wishart-1
has code name: Wishart1.degreesOfFreedom
has Latex symbol: n
parameter has definition expressed in plain text: n > p-1
has short name: degrees of freedom
has short code name: degreesOfFreedom

PDF of Epanechnikov 1
type: probability density function
has name: PDF of Epanechnikov 1
has Latex expression: \frac34 (1-x^2)
has R code expression: 3/4 * (1-x^2)

PMF of Mixture Distribution 1
type: probability mass function
has name: PMF of Mixture Distribution 1
has Latex expression: f(x; \pi, \theta) = \sum_{i=1}^{K} \pi_{i}\; p_i(x; \theta_i) \text{ where } p_i(x; \theta_i) \text{ the PMF of the } i^{th} \text{ component with parameters } \theta_i

relationship between Laplace 2 and Laplace 1 whereby b=1/\\tau
type: Reparameterisation
distribution relationship from: Double Exponential 2
distribution relationship to: Double Exponential 1
distribution relationship definition in Latex: b=1/\tau
distribution relationship pair expression Latex: Laplace2(\mu,\tau) \rightarrow Laplace1(\mu,b)
distribution relationship reference: ProbOnto spec

shape of Nakagami-1
type: mathematical object
has name: shape of Nakagami-1
has code name: Nakagami1.shape
has Latex symbol: m
parameter has definition expressed in plain text: m > 0
has short name: shape
has short code name: shape

mean of Generalized Negative Binomial 1
type: mathematical object
has name: mean of Generalized Negative Binomial 1
has Latex expression: m \theta (1-\theta \beta)^{-1}

scale of Folded-Normal-1
type: mathematical object
has name: scale of Folded-Normal-1
has code name: FoldedNormal1.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \sigma> 0
has short name: scale
has short code name: scale

PDF of Johnson SN 1
type: probability density function
has name: PDF of Johnson SN 1
has Latex expression: \frac{e^{-\frac12 \left( \gamma + \frac{\delta (x-\mu)}{\sigma}\right)^2} \delta}{\sqrt{2\pi}\sigma}
has R code expression: exp(-1/2*(gamma+(delta*(x-mu))/sigma)^2)*delta/(sqrt(2*pi)*sigma)

mode of Binomial 1
type: mathematical object
has name: mode of Binomial 1
has Latex expression: \lfloor (n + 1)p \rfloor \; or \; \lfloor (n + 1)p \rfloor - 1

relationship between Uniform Discrete 1 and Uniform Discrete 2 whereby a = 0, b = n
type: Special case
distribution relationship from: Uniform Discrete 1
distribution relationship to: Uniform Discrete 2
distribution relationship definition in Latex: a = 0, b = n
distribution relationship pair expression Latex: UniformDiscrete1(a,b) \rightarrow UniformDiscrete2(n)
distribution relationship reference: \cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/DiscreteuniformRectangular.pdf}

Conway-Maxwell-Poisson 1
type: probability distribution
has name: Conway-Maxwell-Poisson 1
has code name: ConwayMaxwellPoisson1
distribution has PMF: PMF of Conway-Maxwell-Poisson 1
distribution has CDF: CDF of Conway-Maxwell-Poisson 1
distribution has mean: mean of Conway-Maxwell-Poisson 1
distribution has median: median of Conway-Maxwell-Poisson 1
distribution has variance: variance of Conway-Maxwell-Poisson 1
has variate symbol in Latex: x
has Latex expression of support: x \in \{0,1,2,3,\dots\}
distribution has parameter: rate of decay of Conway-Maxwell-Poisson-1
distribution has parameter: Poisson intensity of Conway-Maxwell-Poisson-1

mode of Frechet 2
type: mathematical object
has name: mode of Frechet 2
has Latex expression: m+\sigma\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}

HF of Standard Power 1
type: hazard function
has name: HF of Standard Power 1
has Latex expression: \frac{\beta x^{\beta-1}}{1 - x^\beta}
has R code expression: beta * x^(beta-1) / (1 - x^beta)

relationship between Logistic 1 and Standard Logistic 1 whereby \\mu = 0, s = 1
type: Special case
distribution relationship from: Logistic 1
distribution relationship to: Standard Logistic 1
distribution relationship definition in Latex: \mu = 0, s = 1
distribution relationship pair expression Latex: Logistic1(\mu,s) \rightarrow StandardLogistic1

mode of Bernoulli 1
type: mathematical object
has name: mode of Bernoulli 1
has Latex expression: \begin{cases} 0 & \text{if } q > p\\ 0, 1 & \text{if } q=p\\ 1 & \text{if } q < p \end{cases}

mean of Student-s-t-distribution-3
type: mathematical object
has name: mean of Student-s-t-distribution-3
has code name: StudentT3.location
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R
has short name: mean
has short code name: location

standard deviation of Log-Normal-7
type: mathematical object
has name: standard deviation of Log-Normal-7
has code name: LogNormal7.stdev
has Latex symbol: \sigma_N
parameter has definition expressed in plain text: \sigma_N > 0
has short name: standard deviation
has short code name: stdev

mode of Lomax 1
type: mathematical object
has name: mode of Lomax 1
has Latex expression: 0

mode of Standard Normal 1
type: mathematical object
has name: mode of Standard Normal 1
has Latex expression: 0

degrees of freedom of Multivariate-Student-T-2
type: mathematical object
has name: degrees of freedom of Multivariate-Student-T-2
has code name: MultivariateStudentT2.degreesOfFreedom
has Latex symbol: k
parameter has definition expressed in plain text: k \ge 2
has short name: degrees of freedom
has short code name: degreesOfFreedom

HF of Standard Triangular 1
type: hazard function
has name: HF of Standard Triangular 1
has Latex expression: \begin{cases} \frac{2(x+1)}{-x^2-2x+1} & \text{ for } -1 < x < 0 \\ \frac2{1-x} & \text{ for } 0 \leq x < 1 \end{cases}
has R code expression: HF1=(2*(x+1))/(-x^2-2*x+1) for -1 < x < 0 HF2=2/(1-x) for 0 \leq x < 1

maximum of Uniform-Discrete-1
type: mathematical object
has name: maximum of Uniform-Discrete-1
has code name: UniformDiscrete1.maximum
has Latex symbol: b
parameter has definition expressed in plain text: b \in \{\dots,-2,-1,0,1,2,3,\dots\}, b \ge a
has short name: maximum
has short code name: maximum

index parameter of Negative-Binomial-3
type: mathematical object
has name: index parameter of Negative-Binomial-3
has code name: NegativeBinomial3.dispersion
has Latex symbol: \phi
parameter has definition expressed in plain text: \phi \in R, \phi > 0
has short name: index parameter
has short code name: dispersion

Binomial distribution with logit parameterisation
type: probability distribution
label: Binomial 2
has name: Binomial 2
has code name: Binomial2
distribution has PMF: PMF of Binomial 2
distribution has CDF: CDF of Binomial 2
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,\dots,n\}
distribution has parameter: logit of success probability in each trial of Binomial-2
distribution has parameter: number of trials of Binomial-2

PDF of Levy 1
type: probability density function
has name: PDF of Levy 1
has Latex expression: \sqrt{\frac{c}{2\pi}} \frac{e^{-\frac{c}{2(x-\mu)}}}{(x-\mu)^{3/2}}
has R code expression: sqrt(c/2/pi) * exp(-c/2/(x-mu)) / (x-mu)^(3/2)

predictor of Ordered-Logistic-1
type: mathematical object
has name: predictor of Ordered-Logistic-1
has code name: OrderedLogistic1.predictor
has Latex symbol: \eta
parameter has definition expressed in plain text: \eta \in R
has short name: predictor
has short code name: predictor

shape2 of Generalized-Gamma-2
type: mathematical object
has name: shape2 of Generalized-Gamma-2
has code name: GeneralizedGamma2.shape2
has Latex symbol: k
parameter has definition expressed in plain text: k > 0
has short name: shape2
has short code name: shape2

scale matrix of Wishart-1
type: mathematical object
has name: scale matrix of Wishart-1
has code name: Wishart1.scaleMatrix
has Latex symbol: V
parameter has definition expressed in plain text: V > 0, p\times p - \text{positive definite matrix}
has short name: scale matrix
has short code name: scaleMatrix

spread of Nakagami-1
type: mathematical object
has name: spread of Nakagami-1
has code name: Nakagami1.spread
has Latex symbol: \Omega
parameter has definition expressed in plain text: \Omega > 0
has short name: spread
has short code name: spread

relationship between Log-Normal 6 and Log-Normal 5 whereby \\mu=\\logm, \\tau= 1/\\log^2 \\sigma_g
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: lognormal
distribution relationship definition in Latex: \mu=\log(m), \tau= 1/\log^2 (\sigma_g)
distribution relationship pair expression Latex: LogNormal6(m,\sigma_g) \rightarrow LogNormal5(\mu,\tau)
distribution relationship reference: ProbOnto spec

CDF of Generalized Negative Binomial 1
type: cumulative density function
has name: CDF of Generalized Negative Binomial 1
has Latex expression: \Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
has R code expression: cumsum(PMF)

relationship between Logistic 2 and Standard Logistic 1 whereby \\mu = 0, \\tau = 1
type: Special case
distribution relationship from: Logistic 2
distribution relationship to: Standard Logistic 1
distribution relationship definition in Latex: \mu = 0, \tau = 1
distribution relationship pair expression Latex: Logistic2(\mu,\tau) \rightarrow StandardLogistic1

interval
type: Class
subclass of: mathematical set

median of Binomial 1
type: mathematical object
has name: median of Binomial 1
has Latex expression: \lfloor np \rfloor \text{ or } \lceil np \rceil

Johnson SN 1
type: probability distribution
has name: Johnson SN 1
has code name: JohnsonSN1
distribution has PDF: PDF of Johnson SN 1
distribution has CDF: CDF of Johnson SN 1
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: location parameter of Johnson-SN-1
distribution has parameter: shape parameter of Johnson-SN-1
distribution has parameter: scale parameter of Johnson-SN-1

CDF of Standard Power 1
type: cumulative density function
has name: CDF of Standard Power 1
has Latex expression: x^\beta
has R code expression: x^beta

median of Frechet 2
type: mathematical object
has name: median of Frechet 2
has Latex expression: m+\frac{\sigma}{\sqrt[\alpha]{\log_e(2)}}

relationship between Normal 3 and Normal 2 whereby \\mu = \\mu, v = 1 / \\tau
type: Reparameterisation
distribution relationship from: Gaussian
distribution relationship to: Normal
distribution relationship definition in Latex: \mu = \mu, v = 1 / \tau
distribution relationship pair expression Latex: Normal3(\mu,\tau) \rightarrow Normal2(\mu,v)
distribution relationship reference: ProbOnto spec

PMF of Conway-Maxwell-Poisson 1
type: probability mass function
has name: PMF of Conway-Maxwell-Poisson 1
has Latex expression: \frac{\lambda^x}{(x!)^\nu}\frac{1}{Z(\lambda,\nu)} \text{ with } Z(\lambda,\nu) = \sum_{j=0}^\infty \frac{\lambda^j}{(j!)^\nu}
has R code expression: lambda^x/(factorial(x))^nu*1/Z(lambda,nu,n); Z(lambda,nu,n): for(i in 0:n) { Z=Z+lambda^i/(factorial(i))^nu }}

GeneralizedPoisson2
type: probability distribution
has name: GeneralizedPoisson2
has code name: GeneralizedPoisson2
distribution has PMF: PMF of GeneralizedPoisson2
distribution has CDF: CDF of GeneralizedPoisson2
distribution has mean: mean of GeneralizedPoisson2
distribution has variance: variance of GeneralizedPoisson2
has variate symbol in Latex: k
has Latex expression of support: k \in \{0,1,2,3,\dots\}
distribution has parameter: mean of GeneralizedPoisson2
distribution has parameter: dispersion of GeneralizedPoisson2

median of Standard Triangular 1
type: mathematical object
has name: median of Standard Triangular 1
has Latex expression: 0

probability of zero of Zero-Inflated-Negative-Binomial-1
type: mathematical object
has name: probability of zero of Zero-Inflated-Negative-Binomial-1
has code name: ZeroInflatedNegativeBinomial1.probabilityOfZero
has Latex symbol: p0
parameter has definition expressed in plain text: 0<p0<1, p0 \in R
has short name: probability of zero
has short code name: probabilityOfZero

mathematical function
type: Class
subclass of: mathematical object

mean of Negative-Binomial-3
type: mathematical object
has name: mean of Negative-Binomial-3
has code name: NegativeBinomial3.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu \in R, \mu > 0
has short name: mean
has short code name: mean

relationship between Negative Binomial 2 and Negative Binomial 3 whereby \\mu=\\lambda, \\phi=1/\\tau
type: Reparameterisation
distribution relationship from: Negative Binomial 2
distribution relationship to: Negative Binomial 3
distribution relationship definition in Latex: \mu=\lambda, \phi=1/\tau
distribution relationship pair expression Latex: NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial3(\mu, \phi)
distribution relationship reference: ProbOnto spec

shape of Generalized-Gamma-3
type: mathematical object
has name: shape of Generalized-Gamma-3
has code name: GeneralizedGamma3.shape2
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0
has short name: shape
has short code name: shape2

mean of x of Log-Normal-7
type: mathematical object
has name: mean of x of Log-Normal-7
has code name: LogNormal7.mean
has Latex symbol: \mu_N
parameter has definition expressed in plain text: \mu_N \in R
has short name: mean of x
has short code name: mean

scale of Student-s-t-distribution-3
type: mathematical object
has name: scale of Student-s-t-distribution-3
has code name: StudentT3.scale
has Latex symbol: \sigma
parameter has definition expressed in plain text: \mu \in R^{+}
has short name: scale
has short code name: scale

mean of Epanechnikov 1
type: mathematical object
has name: mean of Epanechnikov 1
has Latex expression: 0

PMF of Bernoulli 1
type: probability mass function
has name: PMF of Bernoulli 1
has Latex expression: \begin{cases} q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1 \end{cases}
has R code expression: q=(1-p) for k=0 p for k=1

Multinomial 1
type: probability distribution
has name: Multinomial 1
has code name: Multinomial1
distribution has PMF: PMF of Multinomial 1
distribution has mean: mean of Multinomial 1
distribution has variance: variance of Multinomial 1
has variate symbol in Latex: X
has Latex expression of support: X_i \in \{0,\dots,n\}, \Sigma X_i = n
distribution has parameter: number of trials of Multinomial-1
distribution has parameter: event probabilities of Multinomial-1

PMF of Ordered Logistic 1
type: probability mass function
has name: PMF of Ordered Logistic 1
has Latex expression: \begin{cases} 1 - logit^{-1}(\eta - c_1) & \text{for }k=1 \\ logit^{-1}(\eta - c_{k-1}) - logit^{-1}(\eta - c_k) & \text{for }1<k<K \\ logit^{-1}(\eta - c_{k-1}) & \text{for }k=K \end{cases}

degrees of freedom of Chi-1
type: mathematical object
has name: degrees of freedom of Chi-1
has code name: Chi1.degreesOfFreedom
has Latex symbol: k
parameter has definition expressed in plain text: k > 0
has short name: degrees of freedom
has short code name: degreesOfFreedom

CDF of Levy 1
type: cumulative density function
has name: CDF of Levy 1
has Latex expression: \textrm{erfc}\left(\sqrt{\frac{c}{2(x-\mu)}}\right)
has R code expression: erfc(sqrt(c/2/(x-mu)))

variance of Wishart 1
type: mathematical object
has name: variance of Wishart 1
has Latex expression: Var(X_{ij}) = n \left (v_{ij}^2+v_{ii}v_{jj} \right )

scale of Generalized-Negative-Binomial-1
type: mathematical object
has name: scale of Generalized-Negative-Binomial-1
has code name: GeneralizedNegativeBinomial1.theta
has Latex symbol: \theta
parameter has definition expressed in plain text: 0 < \theta < 1
has short name: scale
has short code name: theta

scale parameter of Maxwell-Boltzmann-1
type: mathematical object
has name: scale parameter of Maxwell-Boltzmann-1
has code name: MaxwellBoltzmann1.scale
has Latex symbol: a
parameter has definition expressed in plain text: a > 0
has short name: scale parameter
has short code name: scale

PROB k0001424
type: Transformation
label: relationship between Noncentral F 1 and Noncentral chi-squared 1 whereby F \\sim NoncentralF1\\lambda,m,n \\text{, as } n \\rightarrow \\infty \\text{ then } nF \\text{ approaches a non-central chi-square distribution with } m \\text{ degrees of freedom and non-central parameter } \\lambda
distribution relationship from: Noncentral F 1
distribution relationship to: Noncentral chi-squared 1
distribution relationship definition in Latex: F \sim NoncentralF1(\lambda,m,n) \text{, as } n \rightarrow \infty \text{ then } nF \text{ approaches a non-central chi-square distribution with } m \text{ degrees of freedom and non-central parameter } \lambda
distribution relationship pair expression Latex: NoncentralF1(\lambda,m,n) \rightarrow NoncentralChiSquared1(m,\lambda)
distribution relationship reference: \cite{walck2007handbook}

shape of Normal-inverse-gamma-1
type: mathematical object
has name: shape of Normal-inverse-gamma-1
has code name: NormalInverseGamma1.alpha
has Latex symbol: \alpha
parameter has definition expressed in plain text: \alpha > 0, \alpha \in R
has short name: shape
has short code name: alpha

mixing coefficients of Mixture-Distribution-1
type: mathematical object
has name: mixing coefficients of Mixture-Distribution-1
has code name: MixtureDistribution1.weight
has Latex symbol: \pi_1, \ldots, \pi_k
parameter has definition expressed in plain text: \Sigma_{i=1}^K \pi_i=1; 0\le \pi_i \le 1
has short name: mixing coefficients
has short code name: weight

relationship between Log-Normal 2 and Log-Normal 4 whereby m = \\exp\\mu, cv = \\sqrt{\\expv - 1}
type: Reparameterisation
distribution relationship from: Log-Normal 2
distribution relationship to: lognormal
distribution relationship definition in Latex: m = \exp(\mu), cv = \sqrt{\exp(v) - 1}
distribution relationship pair expression Latex: LogNormal2(\mu,v) \rightarrow LogNormal4(m,cv)
distribution relationship reference: ProbOnto spec

mean of Binomial 1
type: mathematical object
has name: mean of Binomial 1
has Latex expression: np

Arcsine 1
type: probability distribution
has name: Arcsine 1
has code name: Arcsine1
distribution has PDF: PDF of Arcsine 1
distribution has CDF: CDF of Arcsine 1
distribution has SF: SF of Arcsine 1
distribution has HF: HF of Arcsine 1
distribution has mean: mean of Arcsine 1
distribution has median: median of Arcsine 1
distribution has mode: mode of Arcsine 1
distribution has variance: variance of Arcsine 1
has variate symbol in Latex: x
has Latex expression of support: x \in (0,1)

mode of Nakagami 1
type: mathematical object
has name: mode of Nakagami 1
has Latex expression: \frac{\sqrt{2}}{2} \Big(\frac{(2m-1)\Omega}{m} \Big)^{1/2}

mean of Standard Power 1
type: mathematical object
has name: mean of Standard Power 1
has Latex expression: \beta/(\beta+1)

relationship between Generalized Negative Binomial 1 and Binomial 1 whereby \\beta=0 \\text{ and set } m=n, \\theta=p
type: Reparameterisation
distribution relationship from: Generalized Negative Binomial 1
distribution relationship to: Binomial 1
distribution relationship definition in Latex: \beta=0 \text{ and set } m=n, \theta=p
distribution relationship pair expression Latex: GeneralizedNegativeBinomial1(\theta,\beta,m) \rightarrow Binomial1(n,p)
distribution relationship reference: \cite{Consul:2006qf}

mean of Standard-Normal-1
type: mathematical object
has name: mean of Standard-Normal-1
has code name: StandardNormal1.mean
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu=0
has short name: mean
has short code name: mean

relationship between Log-Normal 6 and Log-Normal 1 whereby \\mu=\\logm, \\sigma=\\log\\sigma_g
type: Reparameterisation
distribution relationship from: lognormal
distribution relationship to: Log-Normal 1
distribution relationship definition in Latex: \mu=\log(m), \sigma=\log(\sigma_g)
distribution relationship pair expression Latex: LogNormal6(m,\sigma_g) \rightarrow LogNormal1(\mu,\sigma)
distribution relationship reference: ProbOnto spec

PMF of GeneralizedPoisson2
type: probability mass function
has name: PMF of GeneralizedPoisson2
has Latex expression: \frac{\mu (1-\delta) [\mu(1-\delta)+\delta k]^{k-1}}{k!} e^{-[\mu (1-\delta) + \delta k]}
has R code expression: -(mu*(1-delta)*(mu*(1-delta)+delta*k)^(k-1)) / factorial(k) * exp(-(mu*(1-delta)+delta*k))

mean of Standard Triangular 1
type: mathematical object
has name: mean of Standard Triangular 1
has Latex expression: 0

overdispersion of Zero-Inflated-Negative-Binomial-1
type: mathematical object
has name: overdispersion of Zero-Inflated-Negative-Binomial-1
has code name: ZeroInflatedNegativeBinomial1.overdispersion
has Latex symbol: \tau
has short name: overdispersion
has short code name: overdispersion

matrix
type: Class
subclass of: mathematical object

variance of Log-Normal 7
type: mathematical object
has name: variance of Log-Normal 7
has Latex expression: \sigma_N^2

shape of Generalized-Gamma-3
type: mathematical object
has name: shape of Generalized-Gamma-3
has code name: GeneralizedGamma3.shape1
has Latex symbol: \mu
parameter has definition expressed in plain text: \mu > 0
has short name: shape
has short code name: shape1

CDF of Epanechnikov 1
type: cumulative density function
has name: CDF of Epanechnikov 1
has Latex expression: -\frac14 x^3 + \frac34 x + \frac12
has R code expression: -1/4*x^3 + 3/4*x + 1/2

mean of Bernoulli 1
type: mathematical object
has name: mean of Bernoulli 1
has Latex expression: p

PMF of Multinomial 1
type: probability mass function
has name: PMF of Multinomial 1
has Latex expression: \frac{n!}{x_1!\cdots x_k!} p_1^{x_1} \cdots p_k^{x_k}

variance of Chi 1
type: mathematical object
has name: variance of Chi 1
has Latex expression: k-\mu^2

mode of Wishart 1
type: mathematical object
has name: mode of Wishart 1
has Latex expression: (n-p-1) V \text{ for } n\leq p+1

mean of Levy 1
type: mathematical object
has name: mean of Levy 1
has Latex expression: \infty

t-distribution
type: probability distribution
label: Student's t-distribution 2
has name: Student's t-distribution 2
has code name: StudentT2
distribution has PDF: PDF of Student's t-distribution 2
distribution has mean: mean of Student's t-distribution 2
distribution has mode: mode of Student's t-distribution 2
distribution has variance: variance of Student's t-distribution 2
has variate symbol in Latex: x
has Latex expression of support: x \in (-\infty,+\infty)
distribution has parameter: scale of Student-s-t-distribution-2
distribution has parameter: degrees of freedom of Student-s-t-distribution-2
distribution has parameter: mean of Student-s-t-distribution-2

variance of Generalized Negative Binomial 1
type: mathematical object
has name: variance of Generalized Negative Binomial 1
has Latex expression: m \theta (1-\theta) (1-\theta \beta)^{-3}

scale of Normal-inverse-gamma-1
type: mathematical object
has name: scale of Normal-inverse-gamma-1
has code name: NormalInverseGamma1.beta
has Latex symbol: \beta
parameter has definition expressed in plain text: \beta > 0, \beta \in R
has short name: scale
has short code name: beta

relationship between Noncentral Beta 1 and Beta 1 whereby \\lambda = 0
type: Special case
distribution relationship from: Noncentral Beta 1
distribution relationship to: Beta 1
distribution relationship definition in Latex: \lambda = 0
distribution relationship pair expression Latex: NoncentralBeta1(\lambda, \alpha,\beta) \rightarrow Beta1(\alpha,\beta)
distribution relationship reference: \cite{walck2007handbook}

variance of Maxwell Boltzmann 1
type: mathematical object
has name: variance of Maxwell Boltzmann 1
has Latex expression: \frac{a^2(3 \pi - 8)}{\pi}

CDF of Johnson SN 1
type: cumulative density function
has name: CDF of Johnson SN 1
has Latex expression: \frac12 \text{erfc}\left[ -\frac{\delta + \frac{\delta (x-\mu)}{\sigma}}{\sqrt{2}}\right]
has R code expression: 1/2 *erfc(-(gamma+ (delta*(x-mu)/sigma))/sqrt(2))

SF of Standard Power 1
type: survival function
has name: SF of Standard Power 1
has Latex expression: 1- x^\beta
has R code expression: 1- x^beta

PDF of Arcsine 1
type: probability density function
has name: PDF of Arcsine 1
has Latex expression: \frac{1}{\pi\sqrt{x(1-x)}}
has R code expression: 1/pi/sqrt(x*(1-x))

variance of Nakagami 1
type: mathematical object
has name: variance of Nakagami 1
has Latex expression: \Omega \Big(1-\frac{1}{m}\Big(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\Big)^2 \Big)

variance of Frechet 2
type: mathematical object
has name: variance of Frechet 2
has Latex expression: \begin{cases} \ \sigma^2\left(\Gamma\left(1-\frac{2}{\alpha}\right)- \left(\Gamma\left(1-\frac{1}{\alpha}\right)\right)^2\right) & \text{for } \alpha>2 \\ \ \infty & \text{otherwise} \end{cases}

relationship between Negative Binomial 4 and Negative Binomial 1 whereby p = 1 -p
type: Reparameterisation
distribution relationship from: Negative Binomial 4
distribution relationship to: Negative Binomial 1
distribution relationship definition in Latex: p = 1 -p
distribution relationship pair expression Latex: NegativeBinomial4(r,p) \rightarrow NegativeBinomial1(r,p)
distribution relationship reference: ProbOnto spec