noncentrality parameter of Noncentral-chi-squared-1
\frac{\beta^2}{(\alpha-1)^2(\alpha-2)} \text{ for } \alpha > 2
\sqrt{\frac{\lambda}{2\pi x^3}} \exp\Big(-\frac{\lambda}{2\mu^2 x}(x-\mu)^2\Big)
sqrt(lambda/(2*pi*x^3) ) * exp(-lambda/(2*mu^2 x) * (x-mu)^2)
number of failures of Negative-Binomial-4
{{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}
\frac{\alpha\,x_m^\alpha}{x^{\alpha+1}}\text{ for }x\ge x_m
relationship between Log-Normal 3 and Log-Normal 2 whereby \\mu = \\logm, v = \\sigma^2
LogNormal3(m,\sigma) \rightarrow LogNormal2(\mu,v)
\frac{\pi + 2 \text{ arctan}(\text{sinh}(\pi x))}{2\pi}
\text{Cholesky factor of a symmetric positive-definite matrix with unit diagonal (i.e., a correlation matrix)}
\frac{ \gamma\left(\frac{k}{2},\,\frac{x^2}{2}\right) }{ \Gamma\left(\frac{k}{2}\right) }
\frac12\left[1 + \text{erf}\left( \frac{x-\mu}{\sqrt{1/\tau}\sqrt{2}}\right)\right]
success probability of Negative-Binomial-4
relationship between Log-Normal 5 and Log-Normal 6 whereby m=\\exp\\mu, \\sigma_g= \\exp1/\\sqrt{\\tau}
m=\exp(\mu), \sigma_g= \exp(1/\sqrt{\tau})
LogNormal5(\mu,\tau) \rightarrow LogNormal6(m,\sigma_g)
\frac{1}{x^\alpha H_{n,\alpha}} \\ H_{n,\alpha} = \sum^n_{i=1} (1/i)^\alpha
1 / (x^alpha * Hfunc(n,alpha))\\ Hfunc = function(n,alpha) { Hsum=0; for(i in 1:n) { Hsum = Hsum + (1/i)^alpha } return(Hsum) }
relationship between Laplace 1 and Laplace 2 whereby \\tau=1/b
Laplace1(\mu,b) \rightarrow Laplace2(\mu,\tau)
\begin{cases} \infty & \text{for }\alpha\le 1 \\ \frac{\alpha\,x_m}{\alpha-1} & \text{for }\alpha>1 \end{cases}
degrees of freedom of Noncentral-chi-squared-1
relationship between Log-Logistic 1 and Log-Logistic 2 whereby \\lambda=1/\\alpha, \\kappa=\\beta
LogLogistic1(\alpha,\beta) \rightarrow LogLogistic2(\lambda,\kappa)
(\gamma+\delta\kappa^x) e^{-\gamma x -\delta (\kappa^x - 1)/\log(\kappa)}
(gamma+delta*kappa^x)*exp(-gamma*x -delta*(kappa^x - 1)/log(kappa) )
\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)
pnorm(sqrt(lambda/x) * (x/mu-1)) + exp(2*lambda/mu) * pnorm(-sqrt(lambda/x) * (x/mu+1))
1-\left(\frac{x_m}{x}\right)^\alpha \text{ for } x \ge x_m
\frac{\alpha^2 \beta}{(2+\beta)(1+\beta)^2}
1 - e^{-(\gamma x \log(\kappa) +\delta \kappa^x -\delta)/\log(\kappa)}
1 - exp( -(gamma*x*log(kappa)+delta*kappa^x -delta)/log(kappa) )
\frac{1}{x\sigma\sqrt{2\pi}}\ e^{-\frac{\left(\log x-\mu\right)^2}{2\sigma^2}}
relationship between Generalized Gamma 2 and Exponential 1 whereby k=c=1, a=0, b=1/\\lambda
GeneralizedGamma2(a,b,c,k) \rightarrow Exponential1(\lambda)
\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)}
relationship between Log-Normal 6 and Log-Normal 4 whereby m=m, cv=\\sqrt{\\exp\\!\\big\\log^2\\sigma_g\\big-1}
m=m, cv=\sqrt{\exp\!\big(\log^2(\sigma_g)\big)-1}
LogNormal6(m,\sigma_g) \rightarrow LogNormal4(m,cv)
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)
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) }
relationship between Standard Uniform 1 and Exponential 1 whereby -\\frac{1}{\\lambda} \\logX
StandardUniform1(0,1) \rightarrow Exponential1(\lambda)
\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
X1, X2 \sim Gamma1(k,\theta) \text{ and } Y = X1/(X1+X2) \Rightarrow Y \sim Beta1(\alpha,\beta)
Gamma1(k,\theta) \rightarrow Beta1(\alpha,\beta)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaBeta.pdf}
\frac{m}{m+\beta x} {m+\beta x \choose x} \theta^x (1-\theta)^{m+\beta x-x}
m/(m+beta*x) * choose(m+beta*x,x)*theta^x * (1-theta)^(m+beta*x-x)
relationship between Zero-Inflated Generalized Poisson 1 and Zero-inflated Poisson 1 whereby \\alpha=0, \\lambda=\\mu
ZeroInflatedGeneralizedPoisson1(\mu,\alpha,p0) \rightarrow ZeroInflatedPoisson1(\lambda,\pi)
relationship between Negative Binomial 2 and Negative Binomial 5 whereby \\alpha = 1/\\tau, \\beta = 1 / \\tau \\lambda
\alpha = 1/\tau, \beta = 1 / (\tau \lambda)
NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial5(\alpha, \beta)
\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}
relationship between Negative Binomial 3 and Negative Binomial 4 whereby r = \\phi, p = \\mu / \\phi + \\mu
NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial4(r,p)
relationship between Negative Binomial 4 and Negative Binomial 2 whereby \\tau=1/r, \\lambda = \\frac{rp}{1-p}
NegativeBinomial4(r,p) \rightarrow NegativeBinomial2(\lambda, \tau)
ProbOnto spec \\ \cite{cameron2013regression}
e^{-(\gamma x \log(\kappa) +\delta \kappa^x -\delta)/\log(\kappa)}
exp( -(gamma*x*log(kappa)+delta*kappa^x -delta)/log(kappa) )
{n \choose k}\frac{B(k+\alpha,n-k+\beta)}{B(\alpha,\beta)}
choose(n,k) * beta(k+alpha,n-k+beta) / beta(alpha,beta)
success probability in each trial of Binomial-1
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
\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)
StandardUniform1 \rightarrow LogLogistic2(\lambda,\kappa)
\begin{cases} logit^{-1}(\alpha) & \text{for }k=1 \\ 1 - logit^{-1}(\alpha) & \text{for }k=0 \end{cases}
invLogit = function(x) { exp(x)/(1+exp(x)) } invLogit(alpha1) for k = 1 1-invLogit(alpha1) for k = 0
\begin{cases} \lfloor \frac{p(r-1)}{1-p} \rfloor & \text{for } r > 1 \\ 0 & \text{for } r \leq 1 \end{cases}
relationship between Skew Normal and Standard Normal 1 whereby \\text{If } X \\sim SkewNormal1\\mu,\\sigma,0 \\text{ then } X \\sim StandardNormal1
\text{If } X \sim SkewNormal1(\mu,\sigma,0) \text{ then } X \sim StandardNormal1
SkewNormal1(\mu,\sigma,\alpha) \rightarrow StandardNormal1
\url{https://en.wikipedia.org/wiki/Skew_normal_distribution}\\ \url{http://azzalini.stat.unipd.it/SN/Intro/intro.html}
relationship between Generalized Gamma 1 and Generalized Gamma 3 whereby r=d/p, \\beta=p, \\mu = 1/a
GeneralizedGamma1(a,d,p) \rightarrow GeneralizedGamma3(r,\mu,\beta)
relationship between Generalized Gamma 1 and Gamma 1 whereby p=1, k=d, \\theta=a
GeneralizedGamma1(a,d,p) \rightarrow Gamma1(k,\theta)
max(-1,-\mu/m) < \delta < 1 \text{ with } m (\geq 4) \text { the largest positive integer for which } \mu + m \delta > 0.
\Sigma_{i=1}^x f(i), x \in \{0,1,2,...\} \text{ with } f \text{ the PMF}
\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}
gamma(k + 1/tau)/(factorial(k) * gamma(1/tau)) * 1/(1+tau*lambda)^(1/tau) * (lambda/(1/tau + lambda))^k
\frac12\left(1+ \text{erf}\left[\frac{\gamma + \delta \,\text{arcsin}\left[\frac{x-\mu}{\sigma}\right]}{\sqrt{2}}\right] \right)
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}
X \text{ is logit-normally distributed, then } Y = logit(X)= \log(X/(1-X)) \text{ is normally distributed}
LogitNormal1(\mu,\sigma) \rightarrow Normal1(\mu,\sigma)
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)
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)))) }
relationship between Normal 1 and Log-Normal 1 whereby \\expX
Normal1(\mu,\sigma) \rightarrow LogNormal1(\mu,\sigma)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalLognormal.pdf}\\ \cite{forbes2011statistical}
relationship between Negative Binomial 1 and Normal 1 whereby \\mu = n1-p, n \\rightarrow \\infty
NegativeBinomial1(r,p) \rightarrow Normal1(\mu,\sigma)
\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}
Normal3(\mu,\tau) \rightarrow Normal1(\mu,\sigma)
X(p \times p) - \text{positive-definite matrix}
k \in \{0,1,2,3,\dots\}, \text{ number of failures}
relationship between Negative Binomial 4 and Negative Binomial 3 whereby \\phi = r, \\mu = rp / 1 - p
NegativeBinomial4(r,p) \rightarrow NegativeBinomial3(\mu, \phi)
relationship between Log-Normal 4 and Log-Normal 5 whereby \\mu = \\logm, \\tau = 1 / \\logcv^2 + 1
LogNormal4(m,cv) \rightarrow LogNormal5(\mu,\tau)
\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}
\frac{\sum^x_{i=1} (1/i)^\alpha}{\zeta(\alpha)}
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)) }
\frac{\alpha}{\sigma} \Big(\frac{x}{\sigma}\Big)^{-\alpha-1} \exp\Big(-\Big(\frac{x}{\sigma}\Big)^{-\alpha}\Big)
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.
\text{The Geometric1(p) distribution is a special case of the WeibullDiscrete1} (p, \beta) \text{ distribution when } \beta = 1.
WeibullDiscrete1(p\beta) \rightarrow Geometric1(p)
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
m = \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}, cv = \sigma_N/\mu_N
LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal4(m,cv)
relationship between Rice 1 and Exponential 1 whereby \\text{If } R \\sim Rice10,\\sigma \\text{ then } R^2 \\sim Exponential1\\lambda
\text{If } R \sim Rice1(0,\sigma) \text{ then } R^2 \sim Exponential1(\lambda)
Rice1(\nu,\sigma) \rightarrow Exponential1(\lambda)
\frac{\alpha \pi/\beta}{\sin(\pi/\beta)} \text{ if } \beta>1, \text{ else undefined}
\frac{(\lambda x)^\kappa}{1+(\lambda x)^\kappa}
\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
Exponential1(\lambda) \rightarrow Exponential2(\beta)
\frac{\log(x) - \log(min)}{\log(max) - \log(min)}
(log(x) - log(min)) / (log(max) - log(min))
relationship between Double Poisson 1 and Poisson 1 whereby \\phi=1
DoublePoisson1(\mu,\phi) \rightarrow Poisson1(\lambda)
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}
\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}
LogNormal5(\mu,\tau) \rightarrow LogNormal7(\mu_N,\sigma_N)
relationship between Poisson 1 and Normal 1 whereby \\sigma^2 = \\lambda, \\mu = \\lambda, \\lambda \\rightarrow \\infty
\sigma^2 = \lambda, \mu = \lambda, \lambda \rightarrow \infty
Poisson1(\lambda) \rightarrow Normal1(\mu,\sigma)
\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}
StudentT2(\mu,\tau,k) \rightarrow StudentT3(\nu,\mu,\sigma)
\frac{\pi}{\kappa \lambda \sin(\pi/\kappa)}
\alpha^2 \Big(\frac{2\pi/\beta}{\sin(2\pi/\beta)} - \frac{(\pi/\beta)^2}{\sin^2(\pi/\beta)} \Big), \text{ for } \beta>2
k \in \{0,1,2,3,\dots\} \text{ -- number of successes}
relationship between Gamma 1 and Exponential 1 whereby k=1, \\theta=1/\\lambda
Gamma1(k,\theta) \rightarrow Exponential1(\lambda)
\frac{\zeta(\alpha)\zeta(\alpha-2) - \zeta(\alpha-1)^2}{\zeta(\alpha)^2}
scaling parameter of Scaled-Inverse-Chi-Square
\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}
relationship between Negative Binomial 1 and Poisson 1 whereby \\mu = np, n \\rightarrow \\infty
NegativeBinomial1(r,p) \rightarrow Poisson1(\lambda)
\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}
m = \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}, \sigma = \sqrt{\log(1+\sigma_N^2/\mu_N^2)}
LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal3(m,\sigma)
relationship between Log-Normal 4 and Log-Normal 3 whereby m = m, \\sigma = \\sqrt{\\logcv^2+1}
LogNormal4(m,cv) \rightarrow LogNormal3(m,\sigma)
\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
NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial3(\mu, \phi)
degrees of freedom of Scaled-Inverse-Chi-Square
\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}
(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
\mu = k \theta, \sigma^2 = k^2 \theta, \theta \rightarrow \infty
Gamma1(k,theta) \rightarrow Normal1(\mu,\sigma)
\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
GeneralizedPoisson3(\mu,\alpha) \rightarrow Poisson1(\lambda)
\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}
\mu_N = m \sqrt{cv^2 + 1}, \sigma_N = m \;cv\,\sqrt{cv^2 + 1}
LogNormal4(m,cv) \rightarrow LogNormal7(\mu_N,\sigma_N)
\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} }
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))
\frac{\lambda \kappa(\lambda x)^{\kappa-1}}{(1+(\lambda x)^\kappa)^2}
(lambda*kappa*(lambda*x)^(kappa-1)) / (1+(lambda*x)^kappa)^2
\frac{1}{\Gamma(r)} \gamma\left(r, \mu x\right)
relationship between Exponential 2 and Exponential 1 whereby \\lambda=1/\\beta
Exponential2(\beta) \rightarrow Exponential1(\lambda)
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}
\text{If } X \sim Gamma1(\alpha,\beta) \text{ then } X^{-1} \sim InverseGamma1(\alpha,\beta^{-1})
Gamma1(\alpha,\beta) \rightarrow InverseGamma1(\alpha,\beta)
PDF of Exponentially modified Gaussian 1
\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})
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)
relationship between Frechet 2 and Frechet 1 whereby m=0
Frechet2(\alpha,\sigma,m) \rightarrow Frechet1(\alpha,\sigma)
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}
\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}
Normal1(\mu,\sigma) \rightarrow LogitNormal1(\mu,\sigma)
x \in (-\infty,+\infty), \sigma^2 \in (0,+\infty)
\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)
beta^alpha/gamma(alpha) * x^(-alpha-1) * exp(-beta/x)
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
\mu = \log\Big( \mu_N/\sqrt{1+\sigma_N^2/\mu_N^2} \Big), v = \log(1+\sigma_N^2/\mu_N^2)
LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal2(\mu,v)
relationship between Generalized Gamma 3 and Generalized Gamma 1 whereby a=1/\\mu, d=\\beta r, p=\\beta
GeneralizedGamma3(r,\mu,\beta) \rightarrow GeneralizedGamma1(a,d,p)
\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 \\
relationship between Hypergeometric 1 and Binomial 1 whereby p = K/N, n=n, N \\rightarrow \\infty
Hypergeometric1(N,K,n) \rightarrow Binomial1(n,p)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/HypergeometricBinomial.pdf}
\Psi > 0, \text{positive-definite matrix}
{{\lambda^2 \alpha} \over {(\alpha-1)^2(\alpha-2)}} \text{ for } \alpha > 2;\\ \infty \text{ for } 1 < \alpha \le 2; \\ \text{Otherwise undefined}
\frac{1}{2\,b} \exp \left(-\frac{|x-\mu|}b \right)
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) }
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) }
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}
\mu_N = m\;\exp(\sigma^2/2), \sigma_N = m \;\exp(\sigma^2/2)\sqrt{\exp(\sigma^2)-1}
LogNormal3(m,\sigma) \rightarrow LogNormal7(\mu_N,\sigma_N)
\frac{\Gamma(\alpha, \beta/x)}{\Gamma(\alpha)}
Igamma(alpha, beta/x, lower=F) / gamma(alpha)
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}
\mu = \log\Big(\mu_N/\sqrt{1+\sigma_N^2/\mu_N^2}\Big), \sigma = \sqrt{\log(1+\sigma_N^2/\mu_N^2)}
LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal1(\mu,\sigma)
mean of Exponentially modified Gaussian 1
relationship between Logistic 1 and Logistic 2 whereby \\tau=1/s
Logistic1(\mu,s) \rightarrow Logistic2(\mu,\tau)
{\alpha \over \lambda} \left[{1+ {x-\mu \over \lambda}}\right]^{-(\alpha+1)}
alpha/lambda* (1+ (x-mu)/lambda)^(-(alpha+1))
\frac{\beta}{\alpha-1} \text{ for } \alpha > 1
\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}
relationship between Generalized Gamma 2 and Gamma 1 whereby k=1, a=0 \\text{ and renaming parameters: } c=k, b=\\theta
k=1, a=0 \text{ and renaming parameters: } c=k, b=\theta
GeneralizedGamma2(a,b,c,k) \rightarrow Gamma1(k,\theta)
relationship between Log-Normal 5 and Log-Normal 4 whereby m = \\exp\\mu, cv = \\sqrt{\\exp1/\\tau - 1}
m = \exp(\mu), cv = \sqrt{\exp(1/\tau) - 1}
LogNormal5(\mu,\tau) \rightarrow LogNormal4(m,cv)
CDF of Exponentially modified Gaussian 1
\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
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)))) }
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}
\mu_N = \exp(\mu+v/2), \sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}
LogNormal2(\mu,v) \rightarrow LogNormal7(\mu_N,\sigma_N)
relationship between Erlang 1 and Exponential 2 whereby c=1, b=\\beta
Erlang1(b,c) \rightarrow Exponential2(\beta)
\binom {k+\phi-1}k \Big(\frac{\exp(\eta)}{\exp(\eta) + \phi} \Big)^{k} \Big(\frac{\phi}{\phi + \exp(\eta)} \Big)^{\phi}
choose(k+phi-1,k) * (exp(eta) / (exp(eta) + phi))^k * (phi / (phi + exp(eta)))^phi
1- \left[{1+ {x-\mu \over \lambda}}\right]^{-\alpha}
\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}
StandardUniform1(0,1) \rightarrow ParetoTypeI1(x_m,\alpha)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformPareto.pdf}
relationship between Zero-inflated Poisson 1 and Poisson 1 whereby \\pi=0
ZeroInflatedPoisson1(\lambda,\pi) \rightarrow Poisson1(\lambda)
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
\text{For } \nu=\infty \text{ the distribution of the sum is binomial with parameters } \\ n \text{ and } \lambda/(1+\lambda)
ConwayMaxwellPoisson1(\lambda,\nu) \rightarrow Binomial1(p)
\Phi\Big(\frac{x-\mu}{\sigma}\Big) - 2 T\Big(\frac{x-\mu}{\sigma},\alpha\Big)
Phi((x-mu)/sigma) - 2*T.Owen((x-mu)/sigma,alpha)
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.}
\text{The limiting distribution of a } Zipf1(\alpha, n) \text{ random variable as }\\ n\rightarrow \infty \text{ is the } Zeta1(\alpha) \text{ distribution.}
Zipf1(\alpha,n) \rightarrow Zeta1(\alpha)
shape parameter of Beta-Negative-Binomial1
relationship between Student's t-distribution 3 and Student's t-distribution 1 whereby \\mu=0, \\sigma=1
StudentT3(\nu,\mu,\sigma) \rightarrow StudentT1(\nu)
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
\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)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ChisquareF.pdf}
\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}
relationship between Negative Binomial 1 and Negative Binomial 3 whereby \\phi=r, \\mu=r1-p/p
NegativeBinomial1(r, p) \rightarrow NegativeBinomial3(\mu, \phi)
\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}}
number of successes of Beta-Negative-Binomial1
(beta/alpha)*(x/alpha)^(beta-1) / (1+(x/alpha)^beta)^2
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}
\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}
LogNormal1(\mu,\sigma) \rightarrow LogNormal7(\mu_N,\sigma_N)
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.
StandardTriangular1 \text{ distribution is a special case of the } Triangular1(a,b,c) \text{ distribution } \\ \text{ when } a = -1, b = 1, c = 0.
Triangular1(a,b,c) \rightarrow StandardTriangular1
\mu + \sigma\delta\sqrt{\frac{2}{\pi}} \text{ where } \delta = \frac{\alpha}{\sqrt{1+\alpha^2}}
\frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right)
relationship between Negative Binomial 2 and Negative Binomial 1 whereby r=1/\\tau, p = 1/1+\\tau \\lambda
NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial1(r, p)
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
\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
Normal1(\mu, \sigma) \rightarrow MaxwellBoltzmann1(\sigma)
\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)}
\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}
relationship between Student's t-distribution 2 and Student's t-distribution 1 whereby \\mu=0, \\tau=1
StudentT2(\mu,\tau,k) \rightarrow StudentT1(\nu)
relationship between Kumaraswamy 1 and Exponential 1 whereby \\text{If } X \\sim Kumaraswamy11,b \\text{ then } -1-\\logX \\sim Exponential1b
\text{If } X \sim Kumaraswamy1(1,b) \text{ then } -(1-\log(X)) \sim Exponential1(b)
Kumaraswamy1(a,b) \rightarrow Exponential1(b)
relationship between Log-Normal 3 and Log-Normal 6 whereby m = m, \\sigma_g=\\exp\\sigma
LogNormal3(m,\sigma) \rightarrow LogNormal6(m,\sigma_g)
\frac12\left[1 + \text{erf}\left( \frac{x-\mu}{\sqrt{v}\sqrt{2}}\right)\right]
relationship between Beta 1 and Standard Uniform 1 whereby \\alpha=1, \\beta = 1
Beta1(\alpha,\beta) \rightarrow StandardUniform1
\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}
\begin{cases} \frac{n\beta}{\alpha-1} & \text{if}\ \alpha>1 \\ \infty & \text{otherwise}\ \end{cases}
relationship between Log-Normal 1 and Log-Normal 6 whereby m=\\exp\\mu, \\sigma_g=\\exp\\sigma
LogNormal1(\mu,\sigma) \rightarrow LogNormal6(m,\sigma_g)
relationship between Zipf 1 and Uniform Discrete 1 whereby \\alpha = 0, a = 1, b = n
Zipf1(\alpha,n) \rightarrow UniformDiscrete1(a,b)
\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]
relationship between NoncentralT and Student's t-distribution 1 whereby \\delta=0
NoncentralT1(\delta,n) \rightarrow StudentT1(n)
\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}
\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}}
exp(-1/2 * (gamma+delta *asinh((x-mu)/sigma))^2)* delta / (sqrt(2*pi)*sqrt((x-mu)^2 + sigma^2))
relationship between Log-Normal 2 and Log-Normal 1 whereby \\mu = \\mu, \\sigma = \\sqrt{v}
LogNormal2(\mu,v) \rightarrow LogNormal1(\mu,\sigma)
relationship between Log-Normal 4 and Log-Normal 1 whereby \\mu = \\logm, \\sigma = \\sqrt{\\logcv^2+1}
\mu = \log(m), \sigma = \sqrt{\log(cv^2+1)}
LogNormal4(m,cv) \rightarrow LogNormal1(\mu,\sigma)
parameter has definition expressed in plain text
k/lambda * (x/lambda)^(k-1) * exp(-(x/lambda)^k)
\frac{1}{x\log(\sigma_g)\sqrt{2 \pi}} \exp\Big[ \frac{-[\log(x/m)]^2}{2 \log^2(\sigma_g)}\Big]
relationship between Power-Normal 1 and Log-Normal 1 whereby PowerNormal1 \\text{ distribution converges to LogNormal1 one when } \\lambda \\rightarrow 0
PowerNormal1 \text{ distribution converges to LogNormal1 one when } \lambda \rightarrow 0
PowerNormal1(\lambda,\mu,\sigma) \rightarrow LogNormal1(\mu,\sigma)
\begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases}
\frac12\left[1 + \text{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
relationship between Binomial 1 and Binomial 2 whereby \\alpha = \\logp/1-p
Binomial1(n,p) \rightarrow Binomial2(n,\alpha)
relationship between Gamma 1 and Erlang 1 whereby k \\in N, k=c, \\theta=b
Gamma1(k,\theta) \rightarrow Erlang1(b,c)
\cite{forbes2011statistical} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaErlang.pdf}
\begin{cases} \frac{\nu}{\nu - 2} & \text{for }\nu > 2 \\ \infty & \text{for } 1< \nu \leq 2 \\ undefined & \text{else} \end{cases}
\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)
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.}
\text{As } \kappa \rightarrow 0 \text{ for a Muth1(}\kappa\text{) , the limiting distribution is Exponential2 with mean 1.}
Muth1(\kappa) \rightarrow Exponential2(\beta)
\frac{1}{\Gamma(k)} \gamma\left(k,\, \frac{x}{\theta}\right)
\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}
x/sigma^2*exp(-(x^2+nu^2)/2/sigma^2) * besselI(x*nu/sigma^2, 0, expon.scaled = FALSE)
relationship between Logistic 2 and Logistic 1 whereby s=1/\\tau
Logistic2(\mu,\tau) \rightarrow Logistic1(\mu,s)
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.
\text{The StandardPower1}(\beta) \text{ distribution is a special case of the Power1}(\alpha,\beta) \text{ distribution when } \alpha = 1.
Power1(\alpha,\beta) \rightarrow StandardPower1(\beta)
relationship between Negative Binomial 4 and Negative Binomial 5 whereby \\alpha = r, \\beta = 1-p / p
NegativeBinomial4(r,p) \rightarrow NegativeBinomial5(\alpha, \beta)
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}
\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}
Normal1(\mu,\sigma) \rightarrow FoldedNormal1(\mu,\sigma)
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
Normal1(\mu,\sigma) \rightarrow StandardNormal1(0,1)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalStandardnormalT.pdf}
1 / (gamma(k) * theta^k) * x^(k-1) * exp(-x/theta)
vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-2
\begin{cases} 0 & \text{for }\nu > 1 \\ undefined & \text{else} \end{cases}
relationship between Negative Binomial 3 and Negative Binomial 6 whereby \\eta = \\log\\mu
NegativeBinomial3(\mu,\phi) \rightarrow NegativeBinomial6(\eta,\phi)
boundary separation of Wiener-Diffusion-Model-1
\frac12 \frac{x\sqrt{R^2-x^2}}{\pi R^2} + \frac{\arcsin(\frac{x}{R})}{\pi}
relationship between Uniform 1 and Standard Uniform 1 whereby a = 0, b = 1
Uniform1(a,b) \rightarrow StandardUniform1(0,1)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/UniformStandarduniform.pdf}
relationship between Two-Sided Power 1 and Standard Two-Sided Power 1 whereby a=0, b=1
TwoSidedPower1(a,b,m,n) \rightarrow StandardTwoSidedPower1(m,n)
relationship between Standard Normal 1 and Normal 1 whereby X \\sim StandardNormal1 \\text{ and } Y = \\mu + \\sigma X \\Rightarrow Y \\sim Normal1
X \sim StandardNormal1 \text{ and } Y = \mu + \sigma X \Rightarrow Y \sim Normal1
StandardNormal1(0,1) \rightarrow Normal1(\mu,\sigma)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardnormalNormal.pdf}
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
\text{StandardUniform1 distribution is a special case of the StandardPower1}(\beta) \text{ distribution when } \beta=1
StandardPower1(\beta) \rightarrow StandardUniform1
\frac{\nu \sigma^2}{\nu - 2} \text{ for } \nu>2
relationship between Skew Normal and Chi-squared 1 whereby \\text{If } X \\sim SkewNormal1\\mu,\\sigma,\\alpha \\text{ then } X^2 \\sim ChiSquared11
\text{If } X \sim SkewNormal1(\mu,\sigma,\alpha) \text{ then } X^2 \sim ChiSquared1(1)
SkewNormal1(\mu,\sigma,\alpha) \rightarrow ChiSquared1(k)
lower triangular matrix of Multivariate-Gaussian-Process-Distribution-2
L \in R^{N \times N}, \text{ lower triangular and such that } LL^T \text{ is positive kernel definite}, \text{with } N \in N
relationship between Chi-squared 1 and Exponential 1 whereby k=2 \\text{ and } \\lambda=1/2
ChiSquared1(k) \rightarrow Exponential1(\lambda)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ChisquareExponential.pdf}
relationship between Log-Normal 1 and Log-Normal 4 whereby m = \\exp\\mu, cv = \\sqrt{\\exp\\sigma^2 - 1}
m = \exp(\mu), cv = \sqrt{\exp(\sigma^2) - 1}
LogNormal1(\mu,\sigma) \rightarrow LogNormal4(m,cv)
( 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 }
relationship between Negative Binomial 1 and Geometric 1 whereby n=1
NegativeBinomial1(r,p) \rightarrow Geometric1(p)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalGeometric.pdf}
relationship between Log-Normal 1 and Log-Normal 2 whereby \\mu = \\mu, v = \\sigma^2
LogNormal1(\mu,\sigma) \rightarrow LogNormal2(\mu,v)
initial bias of Wiener-Diffusion-Model-1
\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]
\mu + \beta\,\gamma_E; \text{ with is Euler constant } \gamma_E
relationship between Normal 1 and Normal 3 whereby \\mu = \\mu, \\tau = 1 / \\sigma^2
Normal1(\mu,\sigma) \rightarrow Normal3(\mu,\tau)
\frac{\frac{1}{\sigma} \phi(\frac{x-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}
( 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
Gamma1(k,\theta) \rightarrow Gamma2(r,\mu)
k \in \{0,1,2,3,\dots\} \text{ -- number of failures}
rate of decay of Conway-Maxwell-Poisson-1
degrees of freedom of Student-s-t-distribution-2
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) }
relationship between Zero-Inflated Negative Binomial 1 and Poisson 1 whereby p0=0, \\tau \\rightarrow 0
ZeroInflatedNegativeBinomial1(\lambda,\tau,p0) \rightarrow Poisson1(\lambda)
relationship between Log-Normal 2 and Log-Normal 3 whereby m = \\exp\\mu, \\sigma = \\sqrt{v}
LogNormal2(\mu,v) \rightarrow LogNormal3(m,\sigma)
\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)
relationship between Generalized Negative Binomial 1 and Inverse Binomial 1 whereby \\beta=2, \\theta=1-p
GeneralizedNegativeBinomial1(\theta,\beta,m) \rightarrow InverseBinomial1(k,p)
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
LogNormal2(\mu,v) \rightarrow LogNormal5(\mu,\tau)
\frac{k}\lambda \Big(\frac{x}\lambda\Big)^{k-1}
\frac{1}{\tau} \frac{k}{k-2} \text{ for } k > 2
nondecision time of Wiener-Diffusion-Model-1
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
\text{If } X \sim LogLogistic1(\alpha,\beta) \Rightarrow Y = log(X) \sim Logistic1(\mu,s) \text{ with } \mu=\log(\alpha), s=1/\beta)
LogLogistic1(\alpha,\beta) \rightarrow Logistic1(\mu,s)
covariance matrix of Multivariate-Normal-1
\sigma \sqrt{\tfrac{2}{\pi}} \, e^{(-\mu^2/2\sigma^2)} + \mu \left(1 - 2\,\Phi(\tfrac{-\mu}{\sigma}) \right)
vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-1
\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}
\frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log m}{\sqrt{2}\log(\sigma_g)}\Big]
relationship between Negative Binomial 5 and Negative Binomial 4 whereby r = \\alpha, p = 1 / 1 + \\beta
NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial4(r,p)
\frac{1}{\sigma \sqrt{2 \pi}}\, e^{-\frac{(\operatorname{logit}(x) - \mu)^2}{2\sigma^2}}\frac{1}{x (1-x)}
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
X \sim Frechet2(\alpha,\sigma,m) \text{ with } m=0 \rightarrow X^{-1} \sim Weibull(\lambda=1/\sigma,k=\alpha)
Frechet2(\alpha,\sigma,m) \rightarrow Weibull1(\lambda,k)
\url{https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution} \\ \cite{stan-manual:2015b}
\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]
standard deviation of Truncated-Normal-1
\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]
1/gamma(alpha) * abs(beta/theta) * ((x-a)/theta)^(alpha*beta-1) * exp(-((x-a)/theta)^beta)
relationship between Normal 1 and Standard Normal 1 whereby X \\sim Normal1\\mu,\\sigma; Y = X - \\mu / \\sigma; Y \\sim StandardNormal1
X \sim Normal1(\mu,\sigma); Y = (X - \mu) / \sigma; Y \sim StandardNormal1
Igamma(c, ((x-a)/b)^k, lower=T) / gamma(c)
\frac{\beta x^{\beta-1} \alpha^{-\beta}}{\alpha^\beta - x^\beta}
(beta * x^(beta-1) * alpha^(-beta)) / (alpha^beta - x^beta)
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Frechet1(\alpha,\sigma)
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}
X \sim Hypergeometric1(N, K, n) \Rightarrow Y\sim Normal1(\mu,\sigma) \text{ for large n, but K/N not too small}
Hypergeometric1(N,K,n) \rightarrow Normal1(\mu,\sigma)
\frac12\Big[1 + \operatorname{erf}\Big( \frac{\operatorname{logit}(x)-\mu}{\sqrt{2\sigma^2}}\Big)\Big]
relationship between Gamma 1 and Chi-squared 1 whereby k_{ChiSquared1}=2k, \\theta=2
Gamma1(k,\theta) \rightarrow ChiSquared1(n)
\lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]
logit of probability of success of Bernoulli-2
relationship between GeneralizedPoisson2 and Generalized Poisson 1 whereby \\theta = \\mu 1-\\delta
GeneralizedPoisson2(\mu,\delta) \rightarrow GeneralizedPoisson1(\theta,\delta)
\left(\frac{a-1}{ab-1}\right)^{1/a} \text{ for } a\geq 1, b\geq 1, (a,b)\neq (1,1)
relationship between Geometric 1 and Negative Binomial 1 whereby \\Sigma X iid
Geometric1(p) \rightarrow NegativeBinomial1(r,p)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GeometricPascal.pdf}
relationship between Kumaraswamy 1 and Exponential 1 whereby \\text{If } X \\sim Kumaraswamy1a,1 \\text{ then } -\\logX \\sim Exponential1a
\text{If } X \sim Kumaraswamy1(a,1) \text{ then } -\log(X) \sim Exponential1(a)
Kumaraswamy1(a,b) \rightarrow Exponential1(a)
\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
TruncatedNormal1(\mu,\sigma,a,b) \rightarrow Normal1(\mu,\sigma)
\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 \\
precision matrix of Multivariate-Normal-2
\begin{cases} \lambda \left(\frac{k-1}{k} \right)^{\frac{1}{k}}\, &k>1\\ 0 &k=1\end{cases}
\left \lfloor \frac{(n+1)(K+1)}{N+2} \right \rfloor
relationship between Negative Binomial 1 and Negative Binomial 2 whereby \\tau = 1/r, \\lambda = r1-p/p
NegativeBinomial1(r, p) \rightarrow NegativeBinomial2(\lambda, \tau)
Poisson intensity of Negative-Binomial-2
relationship between Gamma 2 and Gamma 1 whereby k=r, \\theta = 1 / \\mu
Gamma2(r,mu) \rightarrow Gamma1(k,\theta)
relationship between Student's t-distribution 3 and Student's t-distribution 2 whereby \\tau=1/\\sigma^2
StudentT3(\nu,\mu,\sigma) \rightarrow StudentT2(\mu,\tau,k)
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Weibull1(\lambda,k)
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
\text{If } X \sim ScaledInverseChiSquare1(\nu,\sigma) \text{ then } X \sim InverseGamma1(\nu/2,\nu \sigma^2 /2)
ScaledInverseChiSquare1(\nu,\sigma) \rightarrow InverseGamma1(\alpha,\beta)
relationship between Negative Binomial 2 and Negative Binomial 4 whereby r=1/\\tau, p = \\frac{\\tau \\lambda}{1 + \\tau \\lambda}
r=1/\tau, p = \frac{\tau \lambda}{1 + \tau \lambda}
NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial4(r,p)
\mu + \frac{ \phi(\frac{a-\mu}{\sigma}) - \phi(\frac{b-\mu}{\sigma}) }{ \Phi(\frac{b-\mu}{\sigma}) - \Phi(\frac{a-\mu}{\sigma}) } \sigma
relationship between Log-Normal 6 and Log-Normal 2 whereby \\mu=\\logm, v=\\log\\sigma_g^2
LogNormal6(m,\sigma_g) \rightarrow LogNormal2(\mu,v)
x \geq a \text{ if } \theta > 0, x \leq a \text{ if } \theta < 0
(exp(-exp(-(x-mu)/beta)) * exp(-(x-mu)/beta))/beta
relationship between Kumaraswamy 1 and Standard Uniform 1 whereby \\text{If } X \\sim Kumaraswamy11,1 \\text{ then } X \\sim StandardUniform1
\text{If } X \sim Kumaraswamy1(1,1) \text{ then } X \sim StandardUniform1
Kumaraswamy1(a,b) \rightarrow StandardUniform1
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
\text{If } X \sim \text{NegativeBinomial1(n,p) and } p \sim \text{Beta1(}\alpha,\beta \text{) then } X \sim \text{NegativeBinomial1}(\alpha,\beta,n)
BetaNegativeBinomial1(\alpha,\beta,n) \rightarrow NegativeBinomial1(r,p)
relationship between Beta-binomial 1 and Uniform Discrete 2 whereby \\alpha=1, \\beta=1
BetaBinomial1(n,\alpha,\beta) \rightarrow UniformDiscrete2(n)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BetabinomialRectangular.pdf}
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
\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
ConwayMaxwellPoisson1(\lambda,\nu) \rightarrow NegativeBinomial1(r,p)
\frac{k (x-a)^{kc-1}}{b^{kc}\Gamma(c)}\exp\Big[-\Big(\frac{x-a}{b}\Big)^k\Big]
(e^{\kappa x}-\kappa) e^{\left[-\frac{e^{\kappa x}}{\kappa} + \kappa x + \frac1{\kappa} \right]}
(exp(kappa*x)-kappa)*exp( -exp(kappa*x)/x + kappa*x + 1/kappa )
\frac{2}{\sqrt{x(1-x)} (\pi - 2 \arcsin\left(2x -1\right))}
\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}
PMF1=pi + (1-pi)*exp(-lambda) # if k=0 PMF2=(1-pi)*exp(-lambda) * lambda^k/factorial(k) # if k>0
\frac{a p}{x} \left( \frac{(\tfrac{x}{b})^{a p}}{\left((\tfrac{x}{b})^a + 1 \right)^{p+1}} \right)
relationship between Log-Normal 4 and Log-Normal 2 whereby \\mu = \\logm, v = \\logcv^2+1
LogNormal4(m,cv) \rightarrow LogNormal2(\mu,v)
\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}
LogNormal5(\mu,\tau) \rightarrow LogNormal1(\mu,\sigma)
\frac{\gamma(\lfloor k+1 \rfloor,\exp(\alpha))}{\lfloor k \rfloor!}
Igamma(floor(k+1), exp(alpha), lower=F) / factorial(floor(k))
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}
\mu=0, \sigma=\sqrt{\pi} / (\theta \sqrt{2})
HalfNormal2(\mu,\sigma) \rightarrow HalfNormal1(\theta)
E([x=i]) = p_i, \text{this is the mean of the Iverson bracket } [x=i] \text{ and not the mean of } x
\frac{2}{\pi}\arcsin\left(\sqrt x \right)
x \in [0,+\infty) \text{ (Forbes)}, x \in (0,+\infty) \text{ (Leemis)}
\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
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)) } }
relationship between Generalized Poisson 1 and GeneralizedPoisson2 whereby \\mu = \\theta / 1-\\delta
GeneralizedPoisson1(\theta,\delta) \rightarrow GeneralizedPoisson2(\mu,\delta)
\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}}
1-e^{\left[-\frac{e^{\kappa x}}{\kappa} + \kappa x + \frac1{\kappa} \right]}
1-exp( -exp(kappa*x)/x + kappa*x + 1/kappa )
\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]
1/(sqrt(2*pi))* exp( -(sqrt(x/beta) - sqrt(beta/x))^2 / (2*gamma^2) ) * ( sqrt(x/beta) + sqrt(beta/x) ) / (2*gamma*x)
\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]
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) }
\alpha_1, \cdots, \alpha_K, \alpha_i > 0
\exp\left(-\eta\left(e^{bx}-1 \right)\right)
relationship between Log-Normal 4 and Log-Normal 6 whereby m = m, \\sigma_g=\\exp\\!\\big\\sqrt{\\logcv^2+1}\\big
m = m, \sigma_g=\exp\!\big(\sqrt{\log(cv^2+1)}\big)
LogNormal4(m,cv) \rightarrow LogNormal6(m,\sigma_g)
relationship between Trapezoidal 1 and Triangular 1 whereby b=c
Trapezoidal1(a,b,c,d) \rightarrow Triangular1(a,b,c)
\frac1{k!}\exp\big(k \alpha - \exp(\alpha)\big)
1/factorial(k) * exp(k*alpha - exp(alpha))
\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}
relationship between Zero-Inflated Negative Binomial 1 and Zero-inflated Poisson 1 whereby \\tau \\rightarrow 0
ZeroInflatedNegativeBinomial1(\lambda,\tau,p0) \rightarrow ZeroInflatedPoisson1(\lambda,\pi)
relationship between Generalized Negative Binomial 1 and Negative Binomial 4 whereby \\beta=1 \\text{ and set } m=r, \\theta=p
GeneralizedNegativeBinomial1(\theta,\beta,m) \rightarrow NegativeBinomial4(r,p)
relationship between Generalized Gamma 2 and Chi-squared 1 whereby a=0, b=2, c=k_{ChiSquare1}/2,k=1
GeneralizedGamma2(a,b,c,k) \rightarrow ChiSquared1(k)
\frac{\exp(\kappa \cos(x-\mu))}{2\pi I_0(\kappa)}
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
LogLogistic2(\lambda,\kappa) \rightarrow LogLogistic1(\alpha,\beta)
dispersion of Zero-Inflated-Generalized-Poisson-1
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}}
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) }
relationship between Log-Normal 3 and Log-Normal 4 whereby m = m, cv = \\sqrt{\\exp\\sigma^2 - 1}
LogNormal3(m,\sigma) \rightarrow LogNormal4(m,cv)
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)
\mu = [\mu_1, \dots, \mu_d]^T, \mu_i \in R
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}
\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}
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/FChisquare.pdf}
degrees of freedom of Inverse-Chi-Square
X(p \times p) - \text{symmetric, positive definite matrix}
x \in \text{any interval of length } 2\pi
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
X \sim InverseGaussian1(\lambda,\mu) \text{ and } Y = \lambda (X-\mu)^2 / (\mu^2 X) \Rightarrow Y \sim ChiSquared1(k)
InverseGaussian1(\lambda,\mu) \rightarrow ChiSquared1(k)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/InversegaussianChisquare.pdf}
F'=\frac{x_1/m}{x_2/n} \text{ with } x_1 \sim ChiSquared1(m), x_2 \sim ChiSquared1(n)
relationship between Negative Binomial 3 and Negative Binomial 5 whereby \\alpha = \\phi, \\beta = \\phi / \\mu
NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial5(\alpha, \beta)
probability of zero of Zero-Inflated-Generalized-Poisson-1
relationship between Poisson 1 and Poisson 2 whereby \\alpha = \\log\\lambda
Poisson1(\lambda) \rightarrow Poisson2(\alpha)
e^{\left[-\frac{e^{\kappa x}}{\kappa} + \kappa x + \frac1{\kappa} \right]}
exp( -exp(kappa*x)/x + kappa*x + 1/kappa )
\frac{\gamma(\lfloor k+1 \rfloor,\lambda)}{\lfloor k \rfloor!}
Igamma(floor(k+1), lambda, lower=F) / factorial(floor(k))
1-\exp\left(-\eta\left(e^{bx}-1 \right)\right)
relationship between Negative Binomial 6 and Negative Binomial 3 whereby \\mu = \\exp\\eta
NegativeBinomial6(\eta,\phi) \rightarrow NegativeBinomial3(\mu,\phi)
relationship between GeneralizedPoisson2 and Poisson 1 whereby \\delta = 0, \\lambda=\\mu
GeneralizedPoisson2(\mu,\delta) \rightarrow Poisson1(\lambda)
relationship between Normal 2 and Normal 1 whereby \\mu = \\mu, \\sigma = \\sqrt{v}
Normal2(\mu,v) \rightarrow Normal1(\mu,\sigma)
\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}
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)))
\lceil\lambda\rceil - 1, \lfloor\lambda\rfloor
probability of extra zeros of Zero-inflated-Poisson-1
relationship between Standard Normal 1 and Chi-squared 1 whereby \\Sigma X^2
StandardNormal1(0,1) \rightarrow ChiSquared1(k)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardnormalChisquare.pdf}
logit of success probability in each trial of Binomial-2
logit of success probability in each trial
relationship between Log-Normal 6 and Log-Normal 3 whereby m = m, \\sigma=\\log\\sigma_g
LogNormal6(m,\sigma_g) \rightarrow LogNormal3(m,\sigma)
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
\text{If } X \sim ScaledInverseChiSquare1(\nu,\sigma) \text{ then } kX \sim ScaledInverseChiSquare1(\nu, k\sigma)
ScaledInverseChiSquare1(\nu,\sigma) \rightarrow ScaledInverseChiSquare1(\nu,\sigma)
\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}
relationship between Poisson 2 and Poisson 1 whereby \\lambda = \\exp\\alpha
Poisson2(\alpha) \rightarrow Poisson1(\lambda)
number of successes of Negative-Binomial-1
1 - \frac{I_1(\kappa)}{I_1(0)} \text{ (circular variance)}
\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)}
Poisson intensity of Zero-inflated-Poisson-1
relationship between Negative Binomial 5 and Negative Binomial 1 whereby r = \\alpha, p = \\beta / 1 + \\beta
NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial1(r,p)
relationship between Binomial 1 and Poisson 1 whereby \\lambda = np, n \\rightarrow \\infty
Binomial1(n,p) \rightarrow Poisson1(\lambda)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialPoisson.pdf}
degrees of freedom of Multivariate-Student-T-1
relationship between Zero-Inflated Generalized Poisson 1 and Generalized Poisson 3 whereby p0=0
ZeroInflatedGeneralizedPoisson1(\mu,\alpha,p0) \rightarrow GeneralizedPoisson3(\mu,\alpha)
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
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
Hypergeometric1(N,K,n) \rightarrow Poisson1(\lambda)
p\times p - \text{symmetric, positive definite matrix}
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
X\sim StudentT1(\nu) \Rightarrow StandardNormal1(0,1) \text{ as } \nu \text{ tends to infinity.}\\ \text{ The approximation is reasonable for } \nu\ge 30
StudentT1(\nu) \rightarrow StandardNormal1(0,1)
noncentrality parameter of Noncentral-F-1
\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
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 }
Igamma(nu/2,1/2/x, lower=F) / gamma(nu/2)
\binom {k+\alpha-1}{\alpha-1} \Big(\frac{\beta}{\beta + 1} \Big)^{\alpha} \Big(\frac{1}{\beta + 1} \Big)^{k}
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
Weibull1(\lambda,k) \rightarrow Rayleigh1(\sigma)
covariance matrix of Multivariate-Student-T-1
\Sigma, \text{ positive-definite real } p\times p \text{ matrix}
invLogit = function(x) { exp(x)/(1+exp(x)) } \\ Rbeta(1-invLogit(alpha), n-k, 1+k)
\frac{{n-1+x \choose x}{B(n+\alpha,\beta+x)}}{B(\alpha,\beta)}
\left(\frac{n}{m}\right)^2 \frac{2}{(n-2)(n-4)}\left[\frac{(\lambda+m)^2}{n-2} +2\lambda + m \right]
\left( 1+{\left(\frac{x}{b}\right)}^{-a} \right)^{-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
\text{If } X\sim StudentT1(\nu) \Rightarrow Y=X^2 \sim F(1,\nu)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TF.pdf}
transforming parameter of Power-Normal-1
relationship between Zero-Inflated Negative Binomial 1 and Negative Binomial 2 whereby p0=0
ZeroInflatedNegativeBinomial1(\lambda,\tau,p0) \rightarrow NegativeBinomial2(\lambda,\tau)
\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
LogNormal1(\mu,\sigma) \rightarrow LogNormal5(\mu,\tau)
\mu = [\mu_1, \dots, \mu_p]^T, \mu_i \in R
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
\text{For } \nu = 1 \text{ the sum has a Poisson distribution with parameter } n\lambda
ConwayMaxwellPoisson1(\lambda,\nu) \rightarrow Poisson1(\lambda)
\binom {k+\phi-1}k \Big(\frac{\phi}{\mu + \phi} \Big)^{\phi} \Big(\frac{\mu}{\mu + \phi} \Big)^{k}
choose(k+phi-1,k) * (phi / (mu + phi))^phi * (mu / (mu + phi))^k
relationship between Generalized Gamma 2 and Weibull 1 whereby c=1, a=0, b=\\lambda
GeneralizedGamma2(a,b,c,k) \rightarrow Weibull1(\lambda,k)
\begin{cases} \frac{\nu}{\nu-2} \Sigma & \text{for }\nu > 2 \\ undefined & \text{else} \end{cases}
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)
location parameter of Asymmetric-Laplace-1
relationship between Binomial 2 and Binomial 1 whereby p = \\exp\\alpha / 1 + \\exp\\alpha
Binomial2(n,\alpha) \rightarrow Binomial1(n,p)
relationship between Log-Normal 5 and Log-Normal 2 whereby \\mu = \\mu, v = 1 / \\tau
LogNormal5(\mu,\tau) \rightarrow LogNormal2(\mu,v)
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow ChiSquared1(k)
relationship between Negative Binomial 5 and Negative Binomial 2 whereby \\lambda = \\alpha / \\beta, \\tau = 1 / \\alpha
\lambda = \alpha / \beta, \tau = 1 / \alpha
NegativeBinomial5(\alpha, \beta) \rightarrow NegativeBinomial2(\lambda, \tau)
{n \choose k}\, \big(logit^{-1}(\alpha)\big)^k \big(1-logit^{-1}(\alpha)\big)^{n-k}
invLogit = function(x) { exp(x)/(1+exp(x)) } choose(n,k)*invLogit(alpha)^k*(1-invLogit(alpha))^(n-k)
\frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\mu}{\sqrt{2}\sigma}\Big]
relationship between Generalized Poisson 1 and Poisson 1 whereby \\delta = 0, \\theta=\\mu
GeneralizedPoisson1(\theta,\delta) \rightarrow Poisson1(\lambda)
1 - \frac{\Gamma(\beta,x/\alpha)}{\Gamma(\beta)}
relationship between Power-Normal 1 and Truncated Normal 1 whereby PowerNormal1 \\text{ converges to } TruncatedNormal1 \\text{ when } \\lambda \\rightarrow 1
PowerNormal1 \text{ converges to } TruncatedNormal1 \text{ when } \lambda \rightarrow 1
PowerNormal1(\lambda,\mu,\sigma) \rightarrow TruncatedNormal1(\mu,\sigma,a,b)
sqrt(n/2)*gamma((n-1)/2) / gamma(n/2)*delta
relationship between Log-Normal 3 and Log-Normal 5 whereby \\mu = \\logm, \\tau = 1 / \\sigma^2
LogNormal3(m,\sigma) \rightarrow LogNormal5(\mu,\tau)
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Chi1(k)
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
Weibull1(\lambda,k) \rightarrow Weibull2(\lambda,v)
\begin{cases} \mu & \text{for }\nu > 1 \\ undefined & \text{else} \end{cases}
\frac{max^2-min^2}{2[\log(max) - \log(min)]} - \Big(\frac{max-min}{\log(max) - \log(min)}\Big)^2
relationship between Binomial 1 and Bernoulli 1 whereby n=1
Binomial1(n,p) \rightarrow Bernoulli1(p)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialBernoulli.pdf}
relationship between Log-Normal 5 and Log-Normal 3 whereby m = \\exp\\mu, \\sigma = 1 / \\sqrt{\\tau}
m = \exp(\mu), \sigma = 1 / \sqrt{\tau}
LogNormal5(\mu,\tau) \rightarrow LogNormal3(m,\sigma)
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
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)
LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal6(m,\sigma_g)
t'=\frac{x}{\sqrt{y/n}} \text{ with } x \sim Normal2(\delta,1), y \sim ChiSquared1(n).
variance of Zero-Inflated Generalized Poisson 1
\frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\mu}{\sqrt{2/\tau}}\Big]
\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
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
\text{If } X \sim StandardUniform1(0,1) \text{ then } m+\sigma(-\log(X))^{-1/\alpha} \sim Frechet2(\alpha, \sigma, m)
StandardUniform1(0,1) \rightarrow Frechet2(\alpha,\sigma,m)
lower triangular matrix of Multivariate-Normal-3
L \in R^{K \times K}, \text{ lower triangular and such that } LL^T \text{ is positive definite}, \text{with } K \in N
\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}
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) }
degrees of freedom of Student-s-t-distribution-1
mean of Zero-Inflated-Generalized-Poisson-1
\frac{1}{\Gamma\left(\frac{k}{2}\right)}\; \gamma\left(\frac{k}{2},\,\frac{x}{2}\right)
\frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log m}{\sqrt{2}\sigma}\Big]
relationship between Weibull 2 and Weibull 1 whereby k=v, \\lambda = \\lambda^{-1/v}
Weibull2(\lambda,v) \rightarrow Weibull1(\lambda,k)
\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}
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)}
\Big[\exp\Big(-\frac{x}{b}\Big)\Big]\Big( \sum^{c-1}_{i=0} \frac{(x/b)^i}{i!}\Big)
i\text{ such that }p_i=\max(p_1, \ldots, p_k)
\frac{\gamma(d/p, (x/a)^p)}{\Gamma(d/p)}
Igamma(d/p, (x/a)^p, lower=T) / gamma(d/p)
PMF of Zero-Inflated Generalized Poisson 1
\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}
relationship between Rice 1 and Rayleigh 1 whereby \\text{If } R \\sim Rice10,\\sigma \\text{ then } R \\sim Rayleigh1\\sigma
\text{If } R \sim Rice1(0,\sigma) \text{ then } R \sim Rayleigh1(\sigma)
Rice1(\nu,\sigma) \rightarrow Rayleigh1(\sigma)
relationship between Zero-Inflated Generalized Poisson 1 and Poisson 1 whereby p0=0, \\alpha=0, \\lambda=\\mu
ZeroInflatedGeneralizedPoisson1(\mu,\alpha,p0) \rightarrow Poisson1(\lambda)
\frac{\tau}{2} \exp \left(-\tau|x-\mu| \right)
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
\frac{\pi -2 \arcsin\left(2x -1\right)}{2\pi}
CDF of Zero-Inflated Generalized Poisson 1
relationship between Log-Normal 1 and Normal 1 whereby \\logX
LogNormal1(\mu,\sigma) \rightarrow Normal1(\mu,\sigma)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalLognormal.pdf}
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}
\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}
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ExponentialF.pdf}
\frac{\lambda \kappa (\lambda x)^{\kappa-1}}{1+(\lambda x)^\kappa}
lambda * kappa * (lambda * x)^(kappa-1) / (1+(lambda * x)^kappa)
mean of Zero-Inflated Generalized Poisson 1
\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}
relationship between Beta 1 and Arcsine 1 whereby \\alpha=1/2, \\beta = 1/2
Beta1(\alpha,\beta) \rightarrow Arcsine1
relationship between Two-Sided Power 1 and Uniform 1 whereby n=1
TwoSidedPower1(a,b,m,n) \rightarrow Uniform1(a,b)
\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
relationship between Pareto Type I and Exponential 1 whereby X \\sim ParetoTypeI1,Y= \\logX/\\lambda \\Rightarrow Y \\sim Exponential1
X \sim ParetoTypeI1,Y= \log(X/\lambda) \Rightarrow Y \sim Exponential1
ParetoTypeI1(x_m,\alpha) \rightarrow Exponential1(\lambda)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ParetoExponential.pdf}
\frac12\left[1 + \text{erf}\left( \frac{x}{\sqrt{2}}\right)\right]
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Rayleigh1(\sigma)
PDF of Multivariate Gaussian Process Distribution 1
\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
Bernoulli2(\alpha) \rightarrow Bernoulli1(p)
relationship between Pareto Type II and Lomax 1 whereby \\mu=0
ParetoTypeII1(\mu,\lambda,\alpha) \rightarrow Lomax1(\lambda,\alpha)
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Normal1(\mu,\sigma)
\sum_{j=0}^\infty \frac{j^2\lambda^j}{(j!)^\nu Z(\lambda, \nu)} - \text{Mean}^2
{\alpha \over \lambda} \left[{1+ {x \over \lambda}}\right]^{-(\alpha+1)}
alpha/lambda* (1+ x/lambda)^(-(alpha+1))
relationship between Two-Sided Power 1 and Triangular 1 whereby n=2
TwoSidedPower1(a,b,m,n) \rightarrow Triangular1(a,b,m)
\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}
relationship between Sinh-Arcsinh 2 and Normal 1 whereby \\epsilon=0,\\delta=1
SinhArcsinh2(\mu,\sigma,\epsilon,\delta) \rightarrow Normal1(\mu,\sigma)
Multivariate Gaussian Process Distribution 1
relationship between Truncated Normal 1 and Half-normal 1 whereby \\mu=0, a=0, b=\\infty
TruncatedNormal1(\mu,\sigma,a,b) \rightarrow HalfNormal1(\theta)
\url{http://reference.wolfram.com/language/ref/HalfNormalDistribution.html} \\ \cite{forbes2011statistical}
relationship between Negative Binomial 3 and Negative Binomial 1 whereby r=\\phi, p=r/\\mu+r
NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial1(r, p)
a \left(\frac{d-1}{p}\right)^{\frac{1}{p}}, \text{ for } d>1
relationship between Arcsine 2 and Arcsine 1 whereby a=0, b=1
\sqrt{\frac{2}{\pi}} \frac{1}{\sigma} \exp\Big[-\frac{1}{2}\Big(\frac{x-\mu}{\sigma}\Big)^2\Big]
relationship between Log-Normal 2 and Log-Normal 6 whereby m=\\exp\\mu, \\sigma_g=\\exp\\sqrt{v}
LogNormal2(\mu,v) \rightarrow LogNormal6(m,\sigma_g)
relationship between Standard Two-Sided Power 1 and Standard Uniform 1 whereby n=1
StandardTwoSidedPower1(m,n) \rightarrow StandardUniform1
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow InverseChiSquare1(k)
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 }
\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 }
LogNormal6(m,\sigma_g) \rightarrow LogNormal7(\mu_N,\sigma_N)
symmetric, positive definite kernel matrix of Multivariate-Gaussian-Process-Distribution-1
symmetric, positive definite kernel matrix
a^2\left(\frac{\Gamma((d+2)/p)}{\Gamma(d/p)} - \left(\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}\right)^2\right)
1- \left[{1+ {x \over \lambda}}\right]^{-\alpha}
\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}
relationship between Standard Uniform 1 and Uniform 1 whereby X \\sim StandardNormal1 \\text{ and } Y = a + b-a X \\Rightarrow Y \\sim Uniform1
X \sim StandardNormal1 \text{ and } Y = a + (b-a) X \Rightarrow Y \sim Uniform1
StandardUniform1(0,1) \rightarrow Uniform1(a,b)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformUniform.pdf}
\Sigma_{i=1}^x f(i), x \in \{1,2,...\} \text{ with } f \text{ the PMF}
\frac{1}{x\sigma\sqrt{2\pi}}\ e^{-\frac{\left[\log (x/m)\right]^2}{2\sigma^2}}
\sum_{j=0}^\infty \frac{j\lambda^j}{(j!)^\nu Z(\lambda, \nu)}
\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow HalfNormal2(\mu,\sigma)
relationship between Log-Normal 3 and Log-Normal 1 whereby \\mu = \\logm, \\sigma = \\sigma
LogNormal3(m,\sigma) \rightarrow LogNormal1(\mu,\sigma)
relationship between Normal 1 and Levy 1 whereby Y\\,\\sim\\,\\textrm{Normal}\\mu,\\sigma \\Rightarrow {Y-\\mu}^{-2} \\sim\\,\\textrm{Levy}0,1/\\sigma
Y\,\sim\,\textrm{Normal}(\mu,\sigma) \Rightarrow {(Y-\mu)}^{-2} \sim\,\textrm{Levy}(0,1/\sigma)
Normal1(\mu,\sigma) \rightarrow Levy1(\mu,c)
relationship between Sinh-Arcsinh 2 and Standard Normal 1 whereby \\mu=0, \\sigma=1, \\epsilon=0,\\delta=1
SinhArcsinh2(\mu,\sigma,\epsilon,\delta) \rightarrow StandardNormal1
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
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)
LogNormal7(\mu_N,\sigma_N) \rightarrow LogNormal5(\mu,\tau)
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
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)
StandardUniform1(0,1) \rightarrow Beta1(alpha,beta)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformBeta.pdf}
parameter has definition expressed in MathML
\begin{cases} x+1 & \text{ for } -1 < x < 0 \\ 1-x & \text{ for } 0 \leq x < 1 \end{cases}
\frac{\lambda}{\alpha -1} \text{ for } \alpha > 1 \text { otherwise undefined }
standard deviation of Exponentially-modified-Gaussian-1
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
\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)
ScaledInverseChiSquare1 \rightarrow InverseChiSquare1
\frac{2\theta}{\pi} e^{-\theta^2 x^2 / \pi}
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
\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)
Normal1(\mu,\sigma) \rightarrow ChiSquared1(n)
\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}
relationship between Bernoulli 1 and Bernoulli 2 whereby \\alpha = \\logp/1-p
Bernoulli1(p) \rightarrow Bernoulli2(\alpha)
category probabilities of Categorical-Nonordered-1
\text{positive-definite, symmetric matrix with unit diagonal (i.e., a correlation matrix)}
noncentrality parameter of Noncentral-Beta-1
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
\text{Gamma1 distribution is a special case of the Amoroso1} (a,\theta,\alpha,\beta) \text{ distribution when} a = 0, \beta=1
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Gamma1(k,\theta)
relationship between Negative Binomial 3 and Negative Binomial 2 whereby \\lambda=\\mu, \\tau=1/\\phi
NegativeBinomial3(\mu, \phi) \rightarrow NegativeBinomial2(\lambda, \tau)
\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
\frac{\theta (\theta+\delta k)^{k-1}\times e^{-\theta - \delta k}}{k!}
(theta*(theta+k*delta)^(k-1) * exp(-theta-k*delta)) / factorial(k)
variance of Exponentially modified Gaussian 1
\sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!} I_x \left(\alpha + j,\beta\right)
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) }
relationship between Generalized Gamma 2 and Generalized Gamma 1 whereby a = 0, kc=d, \\text{ and rename } k=p, b=a
a = 0, kc=d, \text{ and rename } k=p, b=a
GeneralizedGamma2(a,b,c,k) \rightarrow GeneralizedGamma1(a,d,p)
Poisson intensity of Conway-Maxwell-Poisson-1
relationship between Chi 1 and Chi-squared 1 whereby \\text{If }X \\sim Chi1k \\text{ then } X^2 \\sim ChiSquared1k
\text{If }X \sim Chi1(k) \text{ then } X^2 \sim ChiSquared1(k)
x_i = \frac{\alpha_i - 1}{\sum_{i=1}^K\alpha_i - K}, \quad \alpha_i > 1
\frac{1}{x\sqrt{v}\sqrt{2\pi}}\ e^{-\frac{\left(\log x-\mu\right)^2}{2 v}}
distribution relationship pair expression Latex
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
\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
Amoroso1(a,\theta,\alpha,\beta) \rightarrow Levy1(\mu,c)
\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)}
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
relationship between Beta 1 and Normal 1 whereby \\alpha = \\beta, \\beta \\rightarrow \\infty
\alpha = \beta, \beta \rightarrow \infty
Beta1(alpha,beta) \rightarrow Normal1(\mu,\sigma)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BetaNormal.pdf}
relationship between Frechet 1 and Weibull 1 whereby X \\sim Frechet1\\alpha,\\sigma \\rightarrow X^{-1} \\sim Weibull\\lambda=1/\\sigma,k=\\alpha
X \sim Frechet1(\alpha,\sigma) \rightarrow X^{-1} \sim Weibull(\lambda=1/\sigma,k=\alpha)
Frechet1(\alpha,\sigma) \rightarrow Weibull1(\lambda,k)
\frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\mu}{\sqrt{2}\sqrt{var}}\Big]
relationship between Cauchy 1 and Standard Cauchy 1 whereby x0=0, \\gamma=1
Cauchy1(x_0,\gamma) \rightarrow StandardCauchy1
\frac12 + \frac12\,\text{erf}\Big[\frac{\log x-\log m}{\sqrt{2}\sqrt{\log(cv^2+1)}}\Big]
mean of Zero-Inflated-Negative-Binomial-1
relationship between Bernoulli 1 and Binomial 1 whereby \\Sigma X iid
Bernoulli1(p) \rightarrow Binomial1(n,p)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BernoulliBinomial.pdf}
shape2 of Generalized-Negative-Binomial-1
\frac{\alpha}{\sigma} \Big(\frac{x-m}{\sigma}\Big)^{-\alpha-1} \exp\Big(-\Big(\frac{x-m}{\sigma}\Big)^{-\alpha}\Big)
alpha/sigma * ((x-m)/sigma)^(-alpha-1) * exp(-((x-m)/sigma)^(-alpha))
\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)} \Big(\frac{\Omega}{m}\Big)^{\frac12}
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
relationship between Normal 2 and Normal 3 whereby \\mu = \\mu, \\tau = 1 / v
Normal2(\mu,v) \rightarrow Normal3(\mu,\tau)
category probabilities of Categorical-Ordered-1
relationship between Normal 1 and Normal 2 whereby \\mu = \\mu, v = \\sigma^2
Normal1(\mu,\sigma) \rightarrow Normal2(\mu,v)
mean of Zero-Inflated Negative Binomial 1
E[X_i] = \frac{\alpha_i}{\sum_k \alpha_k}
b\eta e^{bx}e^{\eta}\exp\left(-\eta e^{bx} \right)
relationship between Inverse Gaussian 1 and Standard Normal 1 whereby \\lambda \\rightarrow \\infty
InverseGaussian1(\lambda,\mu) \rightarrow StandardNormal1(0,1)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/InversegaussianStandardnormal.pdf}
Poisson intensity of Generalized-Poisson-1
shape1 of Generalized-Negative-Binomial-1
\beta = 0 \text{ or } 1 \le \beta \le \theta^{-1}
\frac{\beta}{\Gamma(r)}\mu^{\beta r}x^{\beta r -1}\exp[-(\mu x)^\beta]
beta / gamma(r) * mu^(beta*r) * x^(beta*r -1) * exp(-(mu*x)^beta)
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
Weibull1(\lambda,k) \rightarrow Exponential1(\lambda_{Exponetial})
x_1, \cdots, x_K \quad\text{where}\quad x_i \in [0,1]\quad and \quad\sum_{i=1}^K x_i = 1
non-standardized Student's t-distribution
\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}
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
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
\text{If } X \sim \text{ StandardUniform1, then } Y = X^{1/\beta} \text{ has the StandardPower}(\beta) \text{ distribution, where } \beta > 1
StandardUniform1 \rightarrow StandardPower1(\beta)
\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]
(mu/(1+alpha*mu))^y *(1+alpha*y)^(y-1)/factorial(y)*exp(-mu*(1+alpha*y)/(1+alpha*mu))
\frac{2m^m}{\Gamma(m)\Omega^m}x^{2m-1} \exp(-\frac{m}{\omega}x^2)
2*m^m / (gamma(m)*Omega^m)*x^(2*m-1)*exp(-m/Omega*x^2)
\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
\prod^K_{i=1} \text{MultivariateNormal1}(x_i|0,w^{-1}_i LL^T)
1 - \frac12 \text{erfc}\left(-\frac{x}{\sqrt{2}}\right)
X(p \times p) - \text{positive definite matrix}
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.
\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).
Binomial1(n,p) \rightarrow Normal1(\mu,\sigma)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialNormal.pdf}
\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
\text{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \frac{x e^{-x^2/\left(2a^2\right)}}{a}
\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)
PMF of Zero-Inflated Negative Binomial 1
\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}
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
NegativeBinomial1(r,p) \rightarrow NegativeBinomial5(\alpha, \beta)
\frac{e^{-\frac12 \left( \gamma + \delta \log\left[\frac{x-\mu}{\sigma}\right]\right)^2} \delta}{\sqrt{2\pi} (x-\mu)}\\
\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}}
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}
\text{If } X \sim \text{StandardUniform1, then } Y = \lfloor 10^X \rfloor \text{ has the Benford1 distribution}
\frac{\sqrt{\frac2\pi}}{e^{\frac{x^2}{2}} \left(2-\text{erfc}\left(\frac{-x}{\sqrt{2}}\right)\right)}
Multivariate Gaussian Process Distribution
Multivariate Gaussian Process Distribution 2
success probability of Negative-Binomial-1
relationship between Negative Binomial 1 and Negative Binomial 4 whereby p = 1 -p
NegativeBinomial1(r,p) \rightarrow NegativeBinomial4(r,p)
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
\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}
degrees of freedom of Student-s-t-distribution-3
relationship between Log-Normal 1 and Log-Normal 3 whereby m = \\exp\\mu, \\sigma = \\sigma
LogNormal1(\mu,\sigma) \rightarrow LogNormal3(m,\sigma)
\begin{cases} 0 & \text{if } q > p\\ 0.5 & \text{if } q=p\\ 1 & \text{if } q<p \end{cases}
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
\frac{x^{\alpha-1}(1-x)^{\beta-1}} {(1-I_x(\alpha,\beta)) B(\alpha,\beta)}
(x^(alpha-1) * (1-x)^(beta-1))/((1-Rbeta(x,alpha,beta))*beta(alpha,beta))
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
Laplace2(\mu,\tau) \rightarrow Laplace1(\mu,b)
\frac{e^{-\frac12 \left( \gamma + \frac{\delta (x-\mu)}{\sigma}\right)^2} \delta}{\sqrt{2\pi}\sigma}
\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
UniformDiscrete1(a,b) \rightarrow UniformDiscrete2(n)
\cite{Leemis:2008tg} \\ \url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/DiscreteuniformRectangular.pdf}
relationship between Logistic 1 and Standard Logistic 1 whereby \\mu = 0, s = 1
Logistic1(\mu,s) \rightarrow StandardLogistic1
\begin{cases} 0 & \text{if } q > p\\ 0, 1 & \text{if } q=p\\ 1 & \text{if } q < p \end{cases}
degrees of freedom of Multivariate-Student-T-2
\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}
Binomial distribution with logit parameterisation
V > 0, p\times p - \text{positive definite matrix}
relationship between Log-Normal 6 and Log-Normal 5 whereby \\mu=\\logm, \\tau= 1/\\log^2 \\sigma_g
LogNormal6(m,\sigma_g) \rightarrow LogNormal5(\mu,\tau)
relationship between Logistic 2 and Standard Logistic 1 whereby \\mu = 0, \\tau = 1
Logistic2(\mu,\tau) \rightarrow StandardLogistic1
\lfloor np \rfloor \text{ or } \lceil np \rceil
relationship between Normal 3 and Normal 2 whereby \\mu = \\mu, v = 1 / \\tau
Normal3(\mu,\tau) \rightarrow Normal2(\mu,v)
\frac{\lambda^x}{(x!)^\nu}\frac{1}{Z(\lambda,\nu)} \text{ with } Z(\lambda,\nu) = \sum_{j=0}^\infty \frac{\lambda^j}{(j!)^\nu}
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 }}
probability of zero of Zero-Inflated-Negative-Binomial-1
relationship between Negative Binomial 2 and Negative Binomial 3 whereby \\mu=\\lambda, \\phi=1/\\tau
NegativeBinomial2(\lambda, \tau) \rightarrow NegativeBinomial3(\mu, \phi)
\begin{cases} q=(1-p) & \text{for }k=0 \\ p & \text{for }k=1 \end{cases}
\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}
Var(X_{ij}) = n \left (v_{ij}^2+v_{ii}v_{jj} \right )
scale of Generalized-Negative-Binomial-1
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
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
NoncentralF1(\lambda,m,n) \rightarrow NoncentralChiSquared1(m,\lambda)
mixing coefficients of Mixture-Distribution-1
relationship between Log-Normal 2 and Log-Normal 4 whereby m = \\exp\\mu, cv = \\sqrt{\\expv - 1}
LogNormal2(\mu,v) \rightarrow LogNormal4(m,cv)
relationship between Generalized Negative Binomial 1 and Binomial 1 whereby \\beta=0 \\text{ and set } m=n, \\theta=p
GeneralizedNegativeBinomial1(\theta,\beta,m) \rightarrow Binomial1(n,p)
relationship between Log-Normal 6 and Log-Normal 1 whereby \\mu=\\logm, \\sigma=\\log\\sigma_g
LogNormal6(m,\sigma_g) \rightarrow LogNormal1(\mu,\sigma)
-(mu*(1-delta)*(mu*(1-delta)+delta*k)^(k-1)) / factorial(k) * exp(-(mu*(1-delta)+delta*k))
overdispersion of Zero-Inflated-Negative-Binomial-1
variance of Generalized Negative Binomial 1
m \theta (1-\theta) (1-\theta \beta)^{-3}
relationship between Noncentral Beta 1 and Beta 1 whereby \\lambda = 0
NoncentralBeta1(\lambda, \alpha,\beta) \rightarrow Beta1(\alpha,\beta)
\frac12 \text{erfc}\left[ -\frac{\delta + \frac{\delta (x-\mu)}{\sigma}}{\sqrt{2}}\right]
1/2 *erfc(-(gamma+ (delta*(x-mu)/sigma))/sqrt(2))
\Omega \Big(1-\frac{1}{m}\Big(\frac{\Gamma(m+\frac{1}{2})}{\Gamma(m)}\Big)^2 \Big)
relationship between Negative Binomial 4 and Negative Binomial 1 whereby p = 1 -p
NegativeBinomial4(r,p) \rightarrow NegativeBinomial1(r,p)