@prefix rdfs: . @prefix owl: . @prefix xsd: . @prefix rdf: . a owl:Ontology ; owl:versionInfo "2.5.0" . a ; rdfs:label "kurtosis of Sinh-Arcsinh-2" ; "kurtosis of Sinh-Arcsinh-2"^^ ; "SinhArcsinh2.kurtosis"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; "kurtosis"^^ ; "kurtosis"^^ . a ; rdfs:label "shape parameter of Standard-Power-1" ; "shape parameter of Standard-Power-1"^^ ; "StandardPower1.shape"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape parameter"^^ ; "shape"^^ . a ; rdfs:label """noncentrality parameter of Noncentral-chi-squared-1""" ; """noncentrality parameter of Noncentral-chi-squared-1"""^^ ; "NoncentralChiSquared1.noncentrality"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; """noncentrality parameter"""^^ ; "noncentrality"^^ . a ; rdfs:label "variance of Inverse-Gamma 1" ; "variance of Inverse-Gamma 1"^^ ; "\\frac{\\beta^2}{(\\alpha-1)^2(\\alpha-2)} \\text{ for } \\alpha > 2"^^ . a ; rdfs:label "mode of Logistic 1" ; "mode of Logistic 1"^^ ; "\\mu"^^ . a ; rdfs:label "mode of Exponential 1" ; "mode of Exponential 1"^^ ; "0"^^ . a ; rdfs:label "variance of Wigner Semicircle 1" ; "variance of Wigner Semicircle 1"^^ ; "R^2/4"^^ . a ; rdfs:label "mean of GeneralizedPoisson2" ; "mean of GeneralizedPoisson2"^^ ; "GeneralizedPoisson2.mean"^^ ; "\\mu"^^ ; "\\mu > 0"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "PDF of LKJ Correlation 2" ; "PDF of LKJ Correlation 2"^^ ; "\\propto |J| \\det(LL^T)^{(\\eta-1)} = \\prod_{k=2}^{K}L_{kk}^{K-k+2\\eta-2}"^^ . a ; rdfs:label "variance of Generalized Poisson 3" ; "variance of Generalized Poisson 3"^^ ; "\\mu (1+\\alpha \\mu)^2"^^ . a ; rdfs:label "PDF of Multivariate Normal 1" ; "PDF of Multivariate Normal 1"^^ ; "(2\\pi)^{-\\frac{k}{2}}|\\Sigma|^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(x-\\mu)'\\Sigma^{-1}(x-\\mu) }"^^ . a ; rdfs:label "PDF of Inverse Gaussian 1" ; "PDF of Inverse Gaussian 1"^^ ; "\\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)"^^ . a ; rdfs:label "PDF of Chi 1" ; "PDF of Chi 1"^^ ; "\\frac{2^{1-k/2} x^{k-1} e^{-x^2 / 2} }{\\Gamma\\left(\\frac{k}{2}\\right)}"^^ ; "2^(1-k/2) * x^(k-1) * exp(-x^2/2) / gamma(k/2)"^^ . a ; rdfs:label "number of failures of Negative-Binomial-4" ; "number of failures of Negative-Binomial-4"^^ ; "NegativeBinomial4.numberOfFailures"^^ ; "r"^^ ; "r > 0, r \\in N"^^ ; "number of failures"^^ ; "numberOfFailures"^^ . a ; rdfs:label "variance of Trapezoidal 1" ; "variance of Trapezoidal 1"^^ ; """\\dfrac{2^2}{72(d+c-a-b)^2} \\big(d^4+2cd^3-3bd^3-3ad^3 \\\\ -3bcd^2 - 3acd^2 + 4b^2d^2 + 4abd^2 \\\\ +4a^2d^2 + 2c^3d - 3bc^2d - 3ac^2d \\\\ +4b^2cd + 4abcd + 4a^2cd - 3b^3d - 3ab^2d \\\\ -3a^2bd - 3a^3d + c^4 - 3bc^3 - 3ac^3 + 4b^2c^2 \\\\ +4abc^2 + 4a^2c^2 - 3b^3c - 3ab^2c - 3a^2bc \\\\ -3a^3c + b^4 + 2ab^3 + 2a^3b + a4 \\big)"""^^ . a ; rdfs:label "PDF of Normal 3" ; "PDF of Normal 3"^^ ; "\\sqrt{\\frac{\\tau}{2 \\pi}} e^{-\\frac{\\tau}{2}(x-\\mu)^2}"^^ ; "sqrt(tau/(2*pi))*exp(-tau/2*(x-mu)^2)"^^ . a ; rdfs:label "PMF of Hypergeometric 1" ; "PMF of Hypergeometric 1"^^ ; "{{{K \\choose k} {{N-K} \\choose {n-k}}}\\over {N \\choose n}}"^^ ; "choose(K,k)*choose(M-K,n-k)/choose(M,n)"^^ . a ; rdfs:label "PDF of Pareto Type I" ; "PDF of Pareto Type I"^^ ; "\\frac{\\alpha\\,x_m^\\alpha}{x^{\\alpha+1}}\\text{ for }x\\ge x_m"^^ ; "(alpha * x_m^alpha) / x^(alpha+1)"^^ . a ; rdfs:label "relationship between Log-Normal 3 and Log-Normal 2 whereby \\\\mu = \\\\logm, v = \\\\sigma^2" ; ; ; "\\mu = \\log(m), v = \\sigma^2"^^ ; "LogNormal3(m,\\sigma) \\rightarrow LogNormal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape of Inverse-Gamma-1" ; "shape of Inverse-Gamma-1"^^ ; "InverseGamma1.shape"^^ ; "\\alpha"^^ ; "\\alpha>0, \\alpha \\in R"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "variance of Noncentral chi-squared 1" ; "variance of Noncentral chi-squared 1"^^ ; "2(n + 2 \\delta)"^^ . a ; rdfs:label "skewness of Sinh-Arcsinh-2" ; "skewness of Sinh-Arcsinh-2"^^ ; "SinhArcsinh2.skewness"^^ ; "\\epsilon"^^ ; "\\epsilon \\in R"^^ ; "skewness"^^ ; "skewness"^^ . a ; rdfs:label "mean of Normal 3" ; "mean of Normal 3"^^ ; "\\mu"^^ . a ; rdfs:label "radius of Wigner-Semicircle-1" ; "radius of Wigner-Semicircle-1"^^ ; "WignerSemicircle1.radius"^^ ; "R"^^ ; "R > 0"^^ ; "radius"^^ ; "radius"^^ . a ; rdfs:label "median of Exponential 1" ; "median of Exponential 1"^^ ; "\\lambda^{-1} \\ln(2)"^^ . a ; rdfs:label "CDF of Hyperbolic secant 1" ; "CDF of Hyperbolic secant 1"^^ ; "\\frac{\\pi + 2 \\text{ arctan}(\\text{sinh}(\\pi x))}{2\\pi}"^^ ; "(pi + 2*arctan(sinh(pi*x)))/(2*pi)"^^ . a ; rdfs:label "median of Logistic 1" ; "median of Logistic 1"^^ ; "\\mu"^^ . a ; rdfs:label "mean of Generalized Poisson 3" ; "mean of Generalized Poisson 3"^^ ; "\\mu"^^ . a ; rdfs:label "LKJ Correlation 2" ; "LKJ Correlation 2"^^ ; "LKJCorrelation2"^^ ; ; "L"^^ ; "\\text{Cholesky factor of a symmetric positive-definite matrix with unit diagonal (i.e., a correlation matrix)}"^^ ; . a ; rdfs:label "upper bound of Two-Sided-Power-1" ; "upper bound of Two-Sided-Power-1"^^ ; "TwoSidedPower1.upperBound"^^ ; "b"^^ ; "b \\in R, b > m"^^ ; "upper bound"^^ ; "upperBound"^^ . a ; rdfs:label "CDF of Chi 1" ; "CDF of Chi 1"^^ ; "\\frac{ \\gamma\\left(\\frac{k}{2},\\,\\frac{x^2}{2}\\right) }{ \\Gamma\\left(\\frac{k}{2}\\right) }"^^ ; "Igamma(k/2,x^2/2,lower=T) / gamma(k/2)"^^ . a ; rdfs:label "Lower bound of Trapezoidal-1" ; "Lower bound of Trapezoidal-1"^^ ; "Trapezoidal1.lowerBound"^^ ; "a"^^ ; "a < d"^^ ; "Lower bound"^^ ; "lowerBound"^^ . a ; rdfs:label "CDF of Normal 3" ; "CDF of Normal 3"^^ ; "\\frac12\\left[1 + \\text{erf}\\left( \\frac{x-\\mu}{\\sqrt{1/\\tau}\\sqrt{2}}\\right)\\right]"^^ ; "1/2*(1+erf((x-mu)/(sqrt(1/tau)*sqrt(2)))) "^^ . a ; rdfs:label "Inverse Gaussian 1" , "Wald"^^ ; "Inverse Gaussian 1"^^ ; "InverseGaussian1"^^ ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "success probability of Negative-Binomial-4" ; "success probability of Negative-Binomial-4"^^ ; "NegativeBinomial4.probability"^^ ; "p"^^ ; "p \\in (0,1)"^^ ; "success probability"^^ ; "probability"^^ . a ; rdfs:label "Pareto Type I" ; "Pareto Type I"^^ ; "ParetoTypeI1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [x_m, +\\infty)"^^ ; , . a ; rdfs:label "CDF of Hypergeometric 1" ; "CDF of Hypergeometric 1"^^ ; "1-{{{n \\choose {k+1}}{{N-n} \\choose {K-k-1}}}\\over {N \\choose K}} \\,_3F_2\\!\\!\\left[\\begin{array}{c}1,\\ k+1-K,\\ k+1-n \\\\ k+2,\\ N+k+2-K-n\\end{array};1\\right]"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "success probability of Geometric-1" ; "success probability of Geometric-1"^^ ; "Geometric1.probability"^^ ; "p"^^ ; "0 < p < 1"^^ ; "success probability"^^ ; "probability"^^ . a ; rdfs:label """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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Makeham 1" ; "Makeham 1"^^ ; "Makeham1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , , . a ; rdfs:label "Benford 1" ; "Benford 1"^^ ; "Benford1"^^ ; ; ; ; ; ; ; ; ; "k"^^ ; "k \\in \\{1,2,...,9\\}"^^ . a ; rdfs:label "median of Normal 3" ; "median of Normal 3"^^ ; "\\mu"^^ . a ; rdfs:label "PMF of Zipf 1" ; "PMF of Zipf 1"^^ ; """\\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) }"""^^ . a ; rdfs:label "mean of Exponential 1" ; "mean of Exponential 1"^^ ; "\\lambda^{-1}"^^ . a ; rdfs:label "variance of Generalized Gamma 2" ; "variance of Generalized Gamma 2"^^ ; "b^2\\{\\Gamma(c+2/k)/\\Gamma(c)-[\\Gamma(c+1/k)/\\Gamma(c)]^2\\}"^^ . a ; rdfs:label "dispersion of Generalized-Poisson-3" ; "dispersion of Generalized-Poisson-3"^^ ; "GeneralizedPoisson3.dispersion"^^ ; "\\alpha"^^ ; "\\alpha > -1, \\alpha \\in R"^^ ; "dispersion"^^ ; "dispersion"^^ . a ; rdfs:label "scale of Inverse-Gamma-1" ; "scale of Inverse-Gamma-1"^^ ; "InverseGamma1.scale"^^ ; "\\beta"^^ ; "\\beta>0, \\beta \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "median of Standard Power 1" ; "median of Standard Power 1"^^ ; "(1/2)^{1/\\beta}"^^ . a ; rdfs:label "shape paremeter of Amoroso-1" ; "shape paremeter of Amoroso-1"^^ ; "Amoroso1.shape2"^^ ; "\\beta"^^ ; "\\beta \\in R"^^ ; "shape paremeter"^^ ; "shape2"^^ . a ; rdfs:label "mean of Hyperbolic secant 1" ; "mean of Hyperbolic secant 1"^^ ; "0"^^ . a ; rdfs:label "mean of Logistic 1" ; "mean of Logistic 1"^^ ; "\\mu"^^ . a ; rdfs:label "mean of Chi 1" ; "mean of Chi 1"^^ ; "\\mu = \\sqrt{2}\\,\\frac{\\Gamma((k+1)/2)}{\\Gamma(k/2)}"^^ . a ; rdfs:label "shape of Inverse-Gaussian-1" ; "shape of Inverse-Gaussian-1"^^ ; "InverseGaussian1.shape"^^ ; "\\lambda"^^ ; "\\lambda > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "Zero-Inflated Generalized Poisson 1" ; "Zero-Inflated Generalized Poisson 1"^^ ; "ZeroInflatedGeneralizedPoisson1"^^ ; ; ; ; ; "y"^^ ; "y \\in \\{0,1,2,3,\\dots\\}"^^ ; , , . a ; rdfs:label "relationship between Laplace 1 and Laplace 2 whereby \\\\tau=1/b" ; ; ; "\\tau=1/b"^^ ; "Laplace1(\\mu,b) \\rightarrow Laplace2(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Pareto Type I" ; "mean of Pareto Type I"^^ ; """\\begin{cases} \\infty & \\text{for }\\alpha\\le 1 \\\\ \\frac{\\alpha\\,x_m}{\\alpha-1} & \\text{for }\\alpha>1 \\end{cases}"""^^ . a ; rdfs:label "mean of Hypergeometric 1" ; "mean of Hypergeometric 1"^^ ; "n {K\\over N}"^^ . a ; rdfs:label "mode of Normal 3" ; "mode of Normal 3"^^ ; "\\mu"^^ . a ; rdfs:label "SF of Exponential 1" ; "SF of Exponential 1"^^ ; "\\exp(-\\lambda x)"^^ ; "exp(-lambda*x)"^^ . a ; rdfs:label "mode of Generalized Gamma 2" ; "mode of Generalized Gamma 2"^^ ; "a+b(c-1/k)^{1/k}, c>1/k"^^ . a ; rdfs:label "degrees of freedom of Noncentral-chi-squared-1" ; "degrees of freedom of Noncentral-chi-squared-1"^^ ; "NoncentralChiSquared1.degreesOfFreedom"^^ ; "n"^^ ; "n \\in N"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "relationship between Log-Logistic 1 and Log-Logistic 2 whereby \\\\lambda=1/\\\\alpha, \\\\kappa=\\\\beta" ; ; ; "\\lambda=1/\\alpha, \\kappa=\\beta"^^ ; "LogLogistic1(\\alpha,\\beta) \\rightarrow LogLogistic2(\\lambda,\\kappa)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Standard Power 1" ; "variance of Standard Power 1"^^ ; "\\frac{\\beta}{(2+\\beta)(1+\\beta)^2}"^^ . a ; rdfs:label "mean of Generalized-Poisson-3" ; "mean of Generalized-Poisson-3"^^ ; "GeneralizedPoisson3.mean"^^ ; "\\mu"^^ ; "\\mu > 0"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "Zipf 1" ; "Zipf 1"^^ ; "Zipf1"^^ ; ; ; ; ; "x"^^ ; "x \\in \\{1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "shape parameter of Amoroso-1" ; "shape parameter of Amoroso-1"^^ ; "Amoroso1.shape1"^^ ; "\\alpha"^^ ; "\\alpha \\in R, \\alpha > 0"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "PDF of Makeham 1" ; "PDF of Makeham 1"^^ ; "(\\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) )"^^ . a ; rdfs:label "median of Hyperbolic secant 1" ; "median of Hyperbolic secant 1"^^ ; "0"^^ . a ; rdfs:label "CDF of Inverse Gaussian 1" ; "CDF of Inverse Gaussian 1"^^ ; "\\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))"^^ . a ; rdfs:label "Bernoulli 1" ; "Bernoulli 1"^^ ; "Bernoulli1"^^ ; ; ; ; ; ; ; "k"^^ ; "k \\in \\{0,1\\}"^^ ; . a ; rdfs:label "mean of Trapezoidal 1" ; "mean of Trapezoidal 1"^^ ; "\\dfrac{2}{6(d+c-a-b)} \\big(d^2+cd+c^2-b^2-ab-a^2)"^^ . a ; rdfs:label "mode of Chi 1" ; "mode of Chi 1"^^ ; "\\sqrt{k-1} \\text{ for }k\\ge 1"^^ . a ; rdfs:label "CDF of Pareto Type I" ; "CDF of Pareto Type I"^^ ; "1-\\left(\\frac{x_m}{x}\\right)^\\alpha \\text{ for } x \\ge x_m"^^ ; "1-(x_m/x)^alpha"^^ . a ; rdfs:label "variance of Power 1" ; "variance of Power 1"^^ ; "\\frac{\\alpha^2 \\beta}{(2+\\beta)(1+\\beta)^2}"^^ . a ; rdfs:label "CDF of Multivariate Normal 2" ; "CDF of Multivariate Normal 2"^^ ; "\\text{no analytic expression}"^^ . a ; rdfs:label "CDF of Makeham 1" ; "CDF of Makeham 1"^^ ; "1 - e^{-(\\gamma x \\log(\\kappa) +\\delta \\kappa^x -\\delta)/\\log(\\kappa)}"^^ ; "1 - exp( -(gamma*x*log(kappa)+delta*kappa^x -delta)/log(kappa) )"^^ . a ; rdfs:label "PDF of Log-Normal 1" ; "PDF of Log-Normal 1"^^ ; "\\frac{1}{x\\sigma\\sqrt{2\\pi}}\\ e^{-\\frac{\\left(\\log x-\\mu\\right)^2}{2\\sigma^2}}"^^ ; "1/(x*sigma*sqrt(2*pi)) * exp((-(log(x)-mu)^2)/(2*sigma^2))"^^ . a ; rdfs:label "PDF of Multivariate Normal 3" ; "PDF of Multivariate Normal 3"^^ ; "(2\\pi)^{-\\frac{k}{2}}(LL^T)^{-\\frac{1}{2}}\\, e^{ -\\frac{1}{2}(x-\\mu)^T(LL^T)^{-1}(x-\\mu) }"^^ . a ; rdfs:label "scale of Sinh-Arcsinh-1" ; "scale of Sinh-Arcsinh-1"^^ ; "SinhArcsinh1.scale"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "CDF of Benford 1" ; "CDF of Benford 1"^^ ; "\\log_{10}\\left(1+x\\right)"^^ ; "log10(1+x)"^^ . a ; rdfs:label "mode of Hyperbolic secant 1" ; "mode of Hyperbolic secant 1"^^ ; "0"^^ . a ; rdfs:label "shape parameter of Johnson-SU-1" ; "shape parameter of Johnson-SU-1"^^ ; "JohnsonSU1.shape1"^^ ; "\\gamma"^^ ; "\\gamma \\in R"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "Generalized Negative Binomial 1" ; "Generalized Negative Binomial 1"^^ ; "GeneralizedNegativeBinomial1"^^ ; ; ; ; ; "x"^^ ; "x \\in \\{0,1,2,3,\\dots\\}"^^ ; , , . a ; rdfs:label "relationship between Generalized Gamma 2 and Exponential 1 whereby k=c=1, a=0, b=1/\\\\lambda" ; ; ; "k=c=1, a=0, b=1/\\lambda"^^ ; "GeneralizedGamma2(a,b,c,k) \\rightarrow Exponential1(\\lambda)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "median of Geometric 1" ; "median of Geometric 1"^^ ; "\\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)}"^^ . a ; rdfs:label "left hand tail of Sinh-Arcsinh-1" ; "left hand tail of Sinh-Arcsinh-1"^^ ; "SinhArcsinh1.skewness"^^ ; "\\nu"^^ ; "\\nu > 0"^^ ; "left hand tail"^^ ; "skewness"^^ . a ; rdfs:label "median of Exponential 2" ; "median of Exponential 2"^^ ; "\\lambda^{-1} \\ln(2)"^^ . a ; rdfs:label "shape of Weibull-2" ; "shape of Weibull-2"^^ ; "Weibull2.shape"^^ ; "v"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Two-Sided Power 1" ; "mean of Two-Sided Power 1"^^ ; "\\frac{b-m+mn+a}{n+1}"^^ . a ; rdfs:label "CDF of Negative Binomial 4" ; "CDF of Negative Binomial 4"^^ ; "1 - I_{p}(k+1,r)"^^ ; "1 - Rbeta(p, k+1, r,lower = T)"^^ . a ; rdfs:label "PDF of Sinh-Arcsinh 2" ; "PDF of Sinh-Arcsinh 2"^^ ; """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) }"""^^ . a ; rdfs:label "variance of Binomial 1" ; "variance of Binomial 1"^^ ; "np(1 - p)"^^ . a ; rdfs:label "mode of YuleSimon1" ; "mode of YuleSimon1"^^ ; "1"^^ . a ; rdfs:label "relationship between Standard Uniform 1 and Exponential 1 whereby -\\\\frac{1}{\\\\lambda} \\\\logX" ; ; ; "-\\frac{1}{\\lambda} \\log(X)"^^ ; "StandardUniform1(0,1) \\rightarrow Exponential1(\\lambda)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformExponentialB.pdf}"""^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "PDF of Multivariate Normal 2" ; "PDF of Multivariate Normal 2"^^ ; "(2\\pi)^{-d/2}|T|^{\\frac{1}{2}}\\, \\exp\\big( -\\frac{1}{2}(x-\\mu)' T (x-\\mu) \\big)"^^ . a ; rdfs:label "HF of Makeham 1" ; "HF of Makeham 1"^^ ; "\\gamma + \\delta \\kappa^x"^^ ; "gamma + delta*kappa^x"^^ . a ; rdfs:label "scale parameter of Power-1" ; "scale parameter of Power-1"^^ ; "Power1.scale"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "Multivariate Normal 3" , "Cholesky Parameterization"^^ , "Multivariate Normal Distribution"^^ ; "Multivariate Normal 3"^^ ; "MultivariateNormal3"^^ ; ; ; ; ; "x"^^ ; "x \\in R^K"^^ ; , . a ; rdfs:label "Log-Normal 1" , "lognormal"^^ , "Galton"^^ ; "Log-Normal 1"^^ ; "LogNormal1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "Beta-binomial 1" ; "Beta-binomial 1"^^ ; "BetaBinomial1"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,\\dots,n\\}"^^ ; , , . a ; rdfs:label "PMF of Benford 1" ; "PMF of Benford 1"^^ ; "\\log_{10}\\left(1+\\frac1x\\right)"^^ ; "log10(1+1/x)"^^ . a ; rdfs:label "shape parameter of Johnson-SU-1" ; "shape parameter of Johnson-SU-1"^^ ; "JohnsonSU1.shape2"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "variance of Hyperbolic secant 1" ; "variance of Hyperbolic secant 1"^^ ; "1"^^ . a ; rdfs:label "lambda of Weibull-2" ; "lambda of Weibull-2"^^ ; "Weibull2.lambda"^^ ; "\\lambda"^^ ; "lambda"^^ ; "lambda"^^ . a ; rdfs:label "PMF of Generalized Negative Binomial 1" ; "PMF of Generalized Negative Binomial 1"^^ ; "\\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) """^^ . a ; rdfs:label "mean of Exponential 2" ; "mean of Exponential 2"^^ ; "\\beta"^^ . a ; rdfs:label "relationship between Zero-Inflated Generalized Poisson 1 and Zero-inflated Poisson 1 whereby \\\\alpha=0, \\\\lambda=\\\\mu" ; ; ; "\\alpha=0, \\lambda=\\mu"^^ ; "ZeroInflatedGeneralizedPoisson1(\\mu,\\alpha,p0) \\rightarrow ZeroInflatedPoisson1(\\lambda,\\pi)"^^ ; "\\cite{famoye2006zero}"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Geometric 1" ; "mean of Geometric 1"^^ ; "\\frac{1-p}{p}"^^ . a ; rdfs:label "mean of Negative Binomial 4" ; "mean of Negative Binomial 4"^^ ; "\\frac{pr}{1-p}"^^ . a ; rdfs:label "SF of Two-Sided Power 1" ; "SF of Two-Sided Power 1"^^ ; """\\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}"""^^ ; """SF1 = function(x,a,b,m,n) { -(-b+a+(x-a)^n*(m-a)^(1-n))/(b-a) } SF2 = function(x,a,b,m,n) { ((b-x)^n*(b-m)^(1-n))/(b-a) }"""^^ . a ; rdfs:label "variance of YuleSimon1" ; "variance of YuleSimon1"^^ ; "\\frac{\\rho^2}{(\\rho-1)^2(\\rho-2)} \\text{ for } \\rho>2"^^ . a ; rdfs:label "relationship between Negative Binomial 3 and Negative Binomial 4 whereby r = \\\\phi, p = \\\\mu / \\\\phi + \\\\mu" ; ; ; "r = \\phi, p = \\mu / (\\phi + \\mu)"^^ ; "NegativeBinomial3(\\mu, \\phi) \\rightarrow NegativeBinomial4(r,p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "number of trials of Binomial-1" ; "number of trials of Binomial-1"^^ ; "Binomial1.numberOfTrials"^^ ; "n"^^ ; "n \\in N, n \\ge 0"^^ ; "number of trials"^^ ; "numberOfTrials"^^ . a ; rdfs:label "relationship between Negative Binomial 4 and Negative Binomial 2 whereby \\\\tau=1/r, \\\\lambda = \\\\frac{rp}{1-p}" ; ; ; "\\tau=1/r, \\lambda = \\frac{rp}{1-p}"^^ ; "NegativeBinomial4(r,p) \\rightarrow NegativeBinomial2(\\lambda, \\tau)"^^ ; """ProbOnto spec \\\\ \\cite{cameron2013regression}"""^^ . a ; rdfs:label "shape parameter of Power-1" ; "shape parameter of Power-1"^^ ; "Power1.shape"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape parameter"^^ ; "shape"^^ . a ; rdfs:label "Multivariate Normal 2" , "Multivariate Gaussian 2"^^ ; "Multivariate Normal 2"^^ ; "MultivariateNormal2"^^ ; ; ; ; ; ; "x"^^ ; "x \\in R^k"^^ ; , . a ; rdfs:label "SF of Makeham 1" ; "SF of Makeham 1"^^ ; "e^{-(\\gamma x \\log(\\kappa) +\\delta \\kappa^x -\\delta)/\\log(\\kappa)}"^^ ; "exp( -(gamma*x*log(kappa)+delta*kappa^x -delta)/log(kappa) )"^^ . a ; rdfs:label "SF of Benford 1" ; "SF of Benford 1"^^ ; "1-\\log_{10}(x)"^^ ; "1-log10(x)"^^ . a ; rdfs:label "mean of Multivariate Normal 3" ; "mean of Multivariate Normal 3"^^ ; "\\mu"^^ . a ; rdfs:label "Negative Binomial 2" ; "Negative Binomial 2"^^ ; "NegativeBinomial2"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "Ordered Logistic 1" ; "Ordered Logistic 1"^^ ; "OrderedLogistic1"^^ ; ; "k"^^ ; "k \\in \\{1,\\dots, K\\}"^^ ; , . a ; rdfs:label "PMF of Beta-binomial 1" ; "PMF of Beta-binomial 1"^^ ; "{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)"^^ . a ; rdfs:label "rate or inverse scale of Exponential-1" ; "rate or inverse scale of Exponential-1"^^ ; "Exponential1.rate"^^ ; "\\lambda"^^ ; "\\lambda > 0"^^ ; "rate or inverse scale"^^ ; "rate"^^ . a ; rdfs:label "scale of Sinh-Arcsinh-2" ; "scale of Sinh-Arcsinh-2"^^ ; "SinhArcsinh2.scale"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "SF of Exponential 2" ; "SF of Exponential 2"^^ ; "e^{-x/\\beta}"^^ ; "exp(-x/beta)"^^ . a ; rdfs:label "location of Sinh-Arcsinh-2" ; "location of Sinh-Arcsinh-2"^^ ; "SinhArcsinh2.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "scale of Logit-Normal-1" ; "scale of Logit-Normal-1"^^ ; "LogitNormal1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0, \\sigma \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "Lower bound of Two-Sided-Power-1" ; "Lower bound of Two-Sided-Power-1"^^ ; "TwoSidedPower1.lowerBound"^^ ; "a"^^ ; "a \\in R"^^ ; "Lower bound"^^ ; "lowerBound"^^ . a ; rdfs:label "success probability in each trial of Binomial-1" ; "success probability in each trial of Binomial-1"^^ ; "Binomial1.probability"^^ ; "p"^^ ; "p \\in [0,1]"^^ ; "success probability in each trial"^^ ; "probability"^^ . a ; rdfs:label "Zeta 1" ; "Zeta 1"^^ ; "Zeta1"^^ ; ; ; ; ; "x"^^ ; "x \\in \\{1,2,3,\\dots\\}"^^ ; . a ; rdfs:label "relationship between Standard Uniform 1 and Log-Logistic 2 whereby \\\\text{If } X \\\\sim StandardUniform1 \\\\text{ and } Y=\\\\frac{1}{\\\\lambda} \\\\Big \\\\frac{1-X}{X} \\\\Big^{1/\\\\kappa} \\\\Rightarrow Y\\\\sim LogLogistic2\\\\lambda,\\\\kappa" ; ; ; "\\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)"^^ . a ; rdfs:label "PMF of Bernoulli 2" ; "PMF of Bernoulli 2"^^ ; """\\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"""^^ . a ; rdfs:label "mode of Negative Binomial 4" ; "mode of Negative Binomial 4"^^ ; """\\begin{cases} \\lfloor \\frac{p(r-1)}{1-p} \\rfloor & \\text{for } r > 1 \\\\ 0 & \\text{for } r \\leq 1 \\end{cases}"""^^ . a ; rdfs:label "shape parameter of Burr-1" ; "shape parameter of Burr-1"^^ ; "Burr1.shape2"^^ ; "k"^^ ; "k > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "variance of Geometric 1" ; "variance of Geometric 1"^^ ; "\\frac{1-p}{p^2}"^^ . a ; rdfs:label "relationship between Generalized Gamma 1 and Generalized Gamma 3 whereby r=d/p, \\\\beta=p, \\\\mu = 1/a" ; ; ; "r=d/p, \\beta=p, \\mu = 1/a"^^ ; "GeneralizedGamma1(a,d,p) \\rightarrow GeneralizedGamma3(r,\\mu,\\beta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "relationship between Generalized Gamma 1 and Gamma 1 whereby p=1, k=d, \\\\theta=a" ; ; ; "p=1, k=d, \\theta=a"^^ ; "GeneralizedGamma1(a,d,p) \\rightarrow Gamma1(k,\\theta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape parameter of YuleSimon1" ; "shape parameter of YuleSimon1"^^ ; "YuleSimon1.shape"^^ ; "\\rho"^^ ; " \\rho > 0"^^ ; "shape parameter"^^ ; "shape"^^ . a ; rdfs:label "mean of Makeham 1" ; "mean of Makeham 1"^^ ; "\\text{mathematically intractable}"^^ . a ; rdfs:label "Maxwell Boltzmann 1" , "Maxwell"^^ ; "Maxwell Boltzmann 1"^^ ; "MaxwellBoltzmann1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a ; rdfs:label "HF of Benford 1" ; "HF of Benford 1"^^ ; "\\frac{\\log_{10}\\left(1+\\frac1x\\right)}{1-\\log_{10}x}"^^ ; "log10(1+1/x) / (1-log10(x))"^^ . a ; rdfs:label "dispersion of GeneralizedPoisson2" ; "dispersion of GeneralizedPoisson2"^^ ; "GeneralizedPoisson2.dispersion"^^ ; "\\delta"^^ ; "max(-1,-\\mu/m) < \\delta < 1 \\text{ with } m (\\geq 4) \\text { the largest positive integer for which } \\mu + m \\delta > 0."^^ ; "dispersion"^^ ; "dispersion"^^ . a ; rdfs:label "CDF of Beta-binomial 1" ; "CDF of Beta-binomial 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "PMF of Negative Binomial 2" ; "PMF of Negative Binomial 2"^^ ; "\\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"^^ . a ; rdfs:label "variance of Exponential 1" ; "variance of Exponential 1"^^ ; "\\lambda^{-2}"^^ . a ; rdfs:label "CDF of Johnson SU 1" ; "CDF of Johnson SU 1"^^ ; "\\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)))"^^ . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Logit-normal_distribution}"^^ . a ; rdfs:label "variance of Two-Sided Power 1" ; "variance of Two-Sided Power 1"^^ ; "\\frac{-2mnb +2m^2n+a^2n+b^2n-2mna+2bm-2ba+2am-2m^2}{(n+1)^2(n+2)}"^^ . a ; rdfs:label "Hypergeometric 1" ; "Hypergeometric 1"^^ ; "Hypergeometric1"^^ ; ; ; ; ; ; "k"^^ ; "k\\, \\in\\, \\left\\{\\max{(0,\\, n+K-N)},\\, \\dots,\\, \\min{(n,\\, K )}\\right\\}"^^ ; , , . a ; rdfs:label "CDF of Sinh-Arcsinh 2" ; "CDF of Sinh-Arcsinh 2"^^ ; """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)))) }"""^^ . a ; rdfs:label "HF of Exponential 2" ; "HF of Exponential 2"^^ ; "1/\\beta"^^ ; "1/beta"^^ . a ; rdfs:label "variance of Negative Binomial 4" ; "variance of Negative Binomial 4"^^ ; "\\frac{pr}{(1-p)^2}"^^ . a ; rdfs:label "mode of Geometric 1" ; "mode of Geometric 1"^^ ; "0"^^ . a ; rdfs:label "relationship between Normal 1 and Log-Normal 1 whereby \\\\expX" ; ; ; "\\exp(X)"^^ ; "Normal1(\\mu,\\sigma) \\rightarrow LogNormal1(\\mu,\\sigma)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalLognormal.pdf}\\\\ \\cite{forbes2011statistical}"""^^ . a ; rdfs:label "right hand tail of Sinh-Arcsinh-1" ; "right hand tail of Sinh-Arcsinh-1"^^ ; "SinhArcsinh1.kurtosis"^^ ; "\\tau"^^ ; "\\tau > 0"^^ ; "right hand tail"^^ ; "kurtosis"^^ . a ; rdfs:label "Bernoulli 2" ; "Bernoulli 2"^^ ; "Bernoulli2"^^ ; ; "k"^^ ; "k \\in \\{0,1\\}"^^ ; . a ; rdfs:label "shape parameter of Burr-1" ; "shape parameter of Burr-1"^^ ; "Burr1.shape1"^^ ; "c"^^ ; "c > 0"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "Gaussian"^^ , "Normal"^^ , "Normal 3" ; "Normal 3"^^ ; "Normal3"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in R"^^ ; , . a ; rdfs:label "Chi 1" ; "Chi 1"^^ ; "Chi1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a ; rdfs:label """relationship between Negative Binomial 1 and Normal 1 whereby \\\\mu = n1-p, n \\\\rightarrow \\\\infty""" ; ; ; """\\mu = n(1-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}"""^^ . a ; rdfs:label """relationship between Normal 3 and Normal 1 whereby \\\\mu = \\\\mu, \\\\sigma = 1 / \\\\sqrt{\\\\tau}""" ; ; ; """\\mu = \\mu, \\sigma = 1 / \\sqrt{\\tau}"""^^ ; "Normal3(\\mu,\\tau) \\rightarrow Normal1(\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Inverse-Wishart 1" ; "Inverse-Wishart 1"^^ ; "InverseWishart1"^^ ; ; ; ; "X"^^ ; "X(p \\times p) - \\text{positive-definite matrix}"^^ ; , . a ; rdfs:label "minimum of Standard-Uniform-1" ; "minimum of Standard-Uniform-1"^^ ; "StandardUniform1.minimum"^^ ; "a"^^ ; "a=0"^^ ; "minimum"^^ ; "minimum"^^ . a ; rdfs:label "variance of Half Cauchy 1" ; "variance of Half Cauchy 1"^^ ; "undefined"^^ . a ; rdfs:label "Geometric 1" ; "Geometric 1"^^ ; "Geometric1"^^ ; ; ; ; ; ; ; ; ; """k """^^ ; "k \\in \\{0,1,2,3,\\dots\\}, \\text{ number of failures}"^^ ; . a ; rdfs:label "median of Log-Logistic 1" ; "median of Log-Logistic 1"^^ ; "\\alpha"^^ . a ; rdfs:label "relationship between Negative Binomial 4 and Negative Binomial 3 whereby \\\\phi = r, \\\\mu = rp / 1 - p" ; ; ; "\\phi = r, \\mu = rp / (1 - p)"^^ ; "NegativeBinomial4(r,p) \\rightarrow NegativeBinomial3(\\mu, \\phi)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label """relationship between Log-Normal 4 and Log-Normal 5 whereby \\\\mu = \\\\logm, \\\\tau = 1 / \\\\logcv^2 + 1""" ; ; ; """\\mu = \\log(m), \\tau = 1 / \\log(cv^2 + 1)"""^^ ; "LogNormal4(m,cv) \\rightarrow LogNormal5(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Two-Sided Power 1" ; "PDF of Two-Sided Power 1"^^ ; """\\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}"""^^ ; """PDF1 = function(x,a,b,m,n) { (n*(x-a)^(n-1))/((b-a)*(m-a)^(n-1)) } PDF2 = function(x,a,b,m,n) { (n*(b-x)^(n-1))/((b-a)*(b-m)^(n-1)) }"""^^ . a ; rdfs:label "PDF of Log-Uniform 1" ; "PDF of Log-Uniform 1"^^ ; "\\frac{1}{x(\\log(max) - \\log(min))}"^^ ; "1/(x*(log(max) - log(min)))"^^ . a ; rdfs:label "CDF of Zeta 1" ; "CDF of Zeta 1"^^ ; "\\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)) }"""^^ . a ; rdfs:label "CDF of Exponential 2" ; "CDF of Exponential 2"^^ ; "1 - \\exp(-x/\\beta)"^^ ; "1 - exp(-x/beta)"^^ . a ; rdfs:label "CDF of Negative Binomial 2" ; "CDF of Negative Binomial 2"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "PDF of Frechet 1" ; "PDF of Frechet 1"^^ ; "\\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)) "^^ . a ; rdfs:label "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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/DiscreteweibullGeometric.pdf}"^^ . a ; rdfs:label "mean of F 1" ; "mean of F 1"^^ ; "n_2/(n_2-2), n_2>2"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Rice_distribution}"^^ . a ; rdfs:label "mean of Log-Logistic 1" ; "mean of Log-Logistic 1"^^ ; "\\frac{\\alpha \\pi/\\beta}{\\sin(\\pi/\\beta)} \\text{ if } \\beta>1, \\text{ else undefined}"^^ . a ; rdfs:label "CDF of Log-Logistic 2" ; "CDF of Log-Logistic 2"^^ ; "\\frac{(\\lambda x)^\\kappa}{1+(\\lambda x)^\\kappa}"^^ ; "(lambda*x)^kappa / (1+(lambda*x)^kappa)"^^ . a ; rdfs:label "median of Makeham 1" ; "median of Makeham 1"^^ ; "\\text{mathematically intractable}"^^ . a ; rdfs:label "variance of Arcsine 2" ; "variance of Arcsine 2"^^ ; "(b-a)^2/8"^^ . a ; rdfs:label "PDF of Inverse-Wishart 1" ; "PDF of Inverse-Wishart 1"^^ ; "\\frac{\\left|\\Psi\\right|^{\\frac{\\nu}{2}}}{2^{\\frac{\\nu p}{2}}\\Gamma_p(\\frac{\\nu}{2})} \\left|X\\right|^{-\\frac{\\nu+p+1}{2}}e^{-\\frac{1}{2}\\text{tr}(\\Psi X^{-1})}"^^ . a ; rdfs:label "mode of Log-Logistic 1" ; "mode of Log-Logistic 1"^^ ; "\\alpha \\Big(\\frac{\\beta-1}{\\beta+1}\\Big)^{1/\\beta} \\text{ if } \\beta>1, 0 \\text{ otherwise}"^^ . a ; rdfs:label "relationship between Exponential 1 and Exponential 2 whereby \\\\beta=1/\\\\lambda" ; ; ; "\\beta=1/\\lambda"^^ ; "Exponential1(\\lambda) \\rightarrow Exponential2(\\beta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "maximum of Standard-Uniform-1" ; "maximum of Standard-Uniform-1"^^ ; "StandardUniform1.maximum"^^ ; "b"^^ ; "b=1"^^ ; "maximum"^^ ; "maximum"^^ . a ; rdfs:label "median of Half Cauchy 1" ; "median of Half Cauchy 1"^^ ; "\\tan(\\pi/4)"^^ . a ; rdfs:label "concentration of Von-Mises-1" ; "concentration of Von-Mises-1"^^ ; "VonMises1.concentration"^^ ; "\\kappa"^^ ; "\\kappa \\in R^+"^^ ; "concentration"^^ ; "concentration"^^ . a ; rdfs:label "CDF of Log-Uniform 1" ; "CDF of Log-Uniform 1"^^ ; "\\frac{\\log(x) - \\log(min)}{\\log(max) - \\log(min)}"^^ ; "(log(x) - log(min)) / (log(max) - log(min))"^^ . a ; rdfs:label "Two-Sided Power 1" , "TSP"^^ ; "Two-Sided Power 1"^^ ; "TwoSidedPower1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (a,b)"^^ ; , , , . a ; rdfs:label "PMF of Zeta 1" ; "PMF of Zeta 1"^^ ; "\\frac{1}{x^\\alpha \\zeta(\\alpha)}"^^ ; "1 / (x^alpha * zeta(alpha))"^^ . a ; rdfs:label "relationship between Double Poisson 1 and Poisson 1 whereby \\\\phi=1" ; ; ; "\\phi=1"^^ ; "DoublePoisson1(\\mu,\\phi) \\rightarrow Poisson1(\\lambda)"^^ ; "\\cite{cameron2013regression}"^^ . a ; rdfs:label "Frechet 1" ; "Frechet 1"^^ ; "Frechet1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in R^+"^^ ; , . a ; rdfs:label "PDF of Exponential 2" ; "PDF of Exponential 2"^^ ; "1/\\beta e^{-x/\\beta}"^^ ; "1/beta*exp(-1/beta*x)"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Negative Binomial 2" ; "mean of Negative Binomial 2"^^ ; "\\lambda"^^ . a ; rdfs:label "location of Normal-inverse-gamma-1" ; "location of Normal-inverse-gamma-1"^^ ; "NormalInverseGamma1.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "mean"^^ . a ; rdfs:label "CDF of F 1" ; "CDF of F 1"^^ ; "I_{\\frac{n_1 x}{n_1 x + n_2}} \\left(\\tfrac{n_1}{2}, \\tfrac{n_2}{2}\\right)"^^ ; "Rbeta(n1*x / (n1*x + n2), n1/2, n2/2)"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "relationship between Student's t-distribution 2 and Student's t-distribution 3 whereby \\\\sigma=1/\\\\sqrt{\\\\tau}" ; ; ; "\\sigma=1/\\sqrt{\\tau}"^^ ; "StudentT2(\\mu,\\tau,k) \\rightarrow StudentT3(\\nu,\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "degrees of freedom of Noncentral-F-1" ; "degrees of freedom of Noncentral-F-1"^^ ; "NoncentralF1.degreesOfFreedom2"^^ ; "n"^^ ; "n \\in N"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom2"^^ . a ; rdfs:label "mean of Log-Logistic 2" ; "mean of Log-Logistic 2"^^ ; "\\frac{\\pi}{\\kappa \\lambda \\sin(\\pi/\\kappa)}"^^ . a ; rdfs:label "lower bound of Arcsine-2" ; "lower bound of Arcsine-2"^^ ; "Arcsine2.lowerBound"^^ ; "a"^^ ; "a \\in R"^^ ; "lower bound"^^ ; "lowerBound"^^ . a ; rdfs:label "mode of Makeham 1" ; "mode of Makeham 1"^^ ; "\\text{mathematically intractable}"^^ . a ; rdfs:label "CDF of Geometric 1" ; "CDF of Geometric 1"^^ ; "1-(1 - p)^{k+1}"^^ ; "1-(1 - p)^(k+1)"^^ . a ; rdfs:label "variance of Log-Logistic 1" ; "variance of Log-Logistic 1"^^ ; "\\alpha^2 \\Big(\\frac{2\\pi/\\beta}{\\sin(2\\pi/\\beta)} - \\frac{(\\pi/\\beta)^2}{\\sin^2(\\pi/\\beta)} \\Big), \\text{ for } \\beta>2"^^ . a ; rdfs:label "mean of Half Cauchy 1" ; "mean of Half Cauchy 1"^^ ; "undefined"^^ . a ; rdfs:label "Gamma 2" ; "Gamma 2"^^ ; "Gamma2"^^ ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "Negative exponential"^^ , "Exponential 2" ; "Exponential 2"^^ ; "Exponential2"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a ; rdfs:label "Negative Binomial 4" ; "Negative Binomial 4"^^ ; "NegativeBinomial4"^^ ; ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\} \\text{ -- number of successes}"^^ ; , . a ; rdfs:label "relationship between Gamma 1 and Exponential 1 whereby k=1, \\\\theta=1/\\\\lambda" ; ; ; "k=1, \\theta=1/\\lambda"^^ ; "Gamma1(k,\\theta) \\rightarrow Exponential1(\\lambda)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "variance of Zeta 1" ; "variance of Zeta 1"^^ ; "\\frac{\\zeta(\\alpha)\\zeta(\\alpha-2) - \\zeta(\\alpha-1)^2}{\\zeta(\\alpha)^2}"^^ . a ; rdfs:label "scaling parameter of Scaled-Inverse-Chi-Square" ; "scaling parameter of Scaled-Inverse-Chi-Square"^^ ; "ScaledInverseChiSquare1.scale"^^ ; "\\sigma"^^ ; "\\sigma \\in R^+"^^ ; "scaling parameter"^^ ; "scale"^^ . a ; rdfs:label "scale of Birnbaum-Saunders-1" ; "scale of Birnbaum-Saunders-1"^^ ; "BirnbaumSaunders1.scale"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "HF of Two-Sided Power 1" ; "HF of Two-Sided Power 1"^^ ; """\\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}"""^^ ; """HF1 = function(x,a,b,m,n) { -(n*(x-a)^(n-1)*(m-a)^(1-n))/(a-b+(m-a)*(x-a)^n*(m-a)^(-n)) } HF2 = function(x,a,b,m,n) { n/(b-x) } """^^ . a ; rdfs:label "relationship between Negative Binomial 1 and Poisson 1 whereby \\\\mu = np, n \\\\rightarrow \\\\infty" ; ; ; "\\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}"""^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label """relationship between Log-Normal 4 and Log-Normal 3 whereby m = m, \\\\sigma = \\\\sqrt{\\\\logcv^2+1}""" ; ; ; """m = m, \\sigma = \\sqrt{\\log(cv^2+1)}"""^^ ; "LogNormal4(m,cv) \\rightarrow LogNormal3(m,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "probability of Inverse-Binomial-1" ; "probability of Inverse-Binomial-1"^^ ; "InverseBinomial1.probability"^^ ; "p"^^ ; "1/2 < p < 1"^^ ; "probability"^^ ; "probability"^^ . a ; rdfs:label "variance of Negative Binomial 2" ; "variance of Negative Binomial 2"^^ ; "\\lambda (1 + \\tau \\lambda)"^^ . a ; rdfs:label "variance of F 1" ; "variance of F 1"^^ ; "\\frac{2n_2^2(n_1+n_2-2)}{n_1(n_2-2)^2(n_2-4)}, n_2>4"^^ . a ; rdfs:label "upper bound of Arcsine-2" ; "upper bound of Arcsine-2"^^ ; "Arcsine2.upperBound"^^ ; "b"^^ ; "b \\in R, b > a"^^ ; "upper bound"^^ ; "upperBound"^^ . a ; rdfs:label "Fisk"^^ , "Log-Logistic 2" ; "Log-Logistic 2"^^ ; "LogLogistic2"^^ ; ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "scale of Log-Logistic-1" ; "scale of Log-Logistic-1"^^ ; "LogLogistic1.scale"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "PMF of Geometric 1" ; "PMF of Geometric 1"^^ ; "(1 - p)^k\\,p"^^ ; "p*(1-p)^k"^^ . a ; rdfs:label "log Poisson intensity of Poisson-2" ; "log Poisson intensity of Poisson-2"^^ ; "Poisson2.logRate"^^ ; "\\alpha"^^ ; "\\alpha \\in R"^^ ; "log Poisson intensity"^^ ; "logRate"^^ . a ; rdfs:label "SF of Half Cauchy 1" ; "SF of Half Cauchy 1"^^ ; "1 - \\frac2\\pi \\arctan(x)"^^ ; "1 - 2/pi * atan(x)"^^ . a ; rdfs:label "mean of Inverse-Wishart 1" ; "mean of Inverse-Wishart 1"^^ ; "\\frac{\\Psi}{\\nu - p - 1} \\text{ for }\\nu > p + 1"^^ . a ; rdfs:label "relationship between Negative Binomial 5 and Negative Binomial 3 whereby \\\\mu = \\\\alpha / \\\\beta, \\\\phi = \\\\alpha" ; ; ; "\\mu = \\alpha / \\beta, \\phi = \\alpha"^^ ; "NegativeBinomial5(\\alpha, \\beta) \\rightarrow NegativeBinomial3(\\mu, \\phi)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Zeta 1" ; "mean of Zeta 1"^^ ; "\\zeta(\\alpha-1)/\\zeta(\\alpha)"^^ . a ; rdfs:label "degrees of freedom of Scaled-Inverse-Chi-Square" ; "degrees of freedom of Scaled-Inverse-Chi-Square"^^ ; "ScaledInverseChiSquare1.degreesOfFreedom"^^ ; "\\nu"^^ ; "\\nu \\in R^+"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "PMF of Negative Binomial 4" ; "PMF of Negative Binomial 4"^^ ; "\\binom {k+r-1}k (1-p)^r p^k"^^ ; "choose(k+r-1,k) * (1-p)^r * p^k"^^ . a ; rdfs:label "shape of Birnbaum-Saunders-1" ; "shape of Birnbaum-Saunders-1"^^ ; "BirnbaumSaunders1.shape"^^ ; "\\gamma"^^ ; "\\gamma > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "CDF of Two-Sided Power 1" ; "CDF of Two-Sided Power 1"^^ ; """\\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}"""^^ ; """CDF1 = function(x,a,b,m,n) { ((x-a)^n*(m-a)^(1-n))/(b-a) } CDF2 = function(x,a,b,m,n) { -(a+b*(b-x)^n*(b-m)^(-n)-b-m*(b-x)^n*(b-m)^(-n))/(b-a) }"""^^ . a ; rdfs:label "Log-Uniform 1" ; "Log-Uniform 1"^^ ; "LogUniform1"^^ ; ; ; ; ; "x"^^ ; "x \\in (min,max)"^^ ; , . a ; rdfs:label "PDF of Gamma 2" ; "PDF of Gamma 2"^^ ; "\\frac{\\mu^r x^{r-1} e^{-\\mu x}}{\\Gamma(r)}"^^ ; "(mu^r * x^(r-1) * exp(-mu*x)) / gamma(r)"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "relationship between Generalized Poisson 3 and Poisson 1 whereby \\\\alpha = 0, \\\\lambda=\\\\mu" ; ; ; "\\alpha = 0, \\lambda=\\mu"^^ ; "GeneralizedPoisson3(\\mu,\\alpha) \\rightarrow Poisson1(\\lambda)"^^ ; """\\cite{hilbe2011negative} \\\\ \\cite{famoye2006zero}"""^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Normal-inverse-gamma 1" ; "PDF of Normal-inverse-gamma 1"^^ ; "\\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))"^^ . a ; rdfs:label "mode of F 1" ; "mode of F 1"^^ ; "\\frac{n_2(n_1-2)}{n_1(n_2+2)}, n_1>2"^^ . a ; rdfs:label "PDF of Log-Logistic 2" ; "PDF of Log-Logistic 2"^^ ; "\\frac{\\lambda \\kappa(\\lambda x)^{\\kappa-1}}{(1+(\\lambda x)^\\kappa)^2}"^^ ; "(lambda*kappa*(lambda*x)^(kappa-1)) / (1+(lambda*x)^kappa)^2"^^ . a ; rdfs:label "Noncentral Beta 1" ; "Noncentral Beta 1"^^ ; "NoncentralBeta1"^^ ; ; ; "x"^^ ; "x \\in [0,1]"^^ ; , , . a ; rdfs:label "CDF of Gamma 2" ; "CDF of Gamma 2"^^ ; "\\frac{1}{\\Gamma(r)} \\gamma\\left(r, \\mu x\\right)"^^ ; "1/gamma(r) * Igamma(r,mu*x,lower=T)"^^ . a ; rdfs:label "Inverse-Gamma 1" ; "Inverse-Gamma 1"^^ ; "InverseGamma1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "noncentrality parameter of Rice-1" ; "noncentrality parameter of Rice-1"^^ ; "Rice1.noncentrality"^^ ; "\\nu"^^ ; "\\nu \\geq 0"^^ ; "noncentrality parameter"^^ ; "noncentrality"^^ . a owl:Class ; rdfs:label "Transformation" ; rdfs:subClassOf . a ; rdfs:label "relationship between Exponential 2 and Exponential 1 whereby \\\\lambda=1/\\\\beta" ; ; ; "\\lambda=1/\\beta"^^ ; "Exponential2(\\beta) \\rightarrow Exponential1(\\lambda)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Trapezoidal 1" , "Piecewise Random"^^ ; "Trapezoidal 1"^^ ; "Trapezoidal1"^^ ; ; ; ; ; "x"^^ ; "x \\in [a,d]"^^ ; , , , . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Inverse-gamma_distribution}"^^ . a ; rdfs:label "mode of Inverse-Wishart 1" ; "mode of Inverse-Wishart 1"^^ ; "\\frac{\\Psi}{\\nu + p + 1}"^^ . a ; rdfs:label "PDF of Exponentially modified Gaussian 1" ; "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)"""^^ . a ; rdfs:label "variance of Poisson 2" ; "variance of Poisson 2"^^ ; "\\exp(\\alpha)"^^ . a ; rdfs:label "Dagum 1" ; "Dagum 1"^^ ; "Dagum1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; , , . a ; rdfs:label "shape of Log-Logistic-1" ; "shape of Log-Logistic-1"^^ ; "LogLogistic1.shape"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "scale of Lomax-1" ; "scale of Lomax-1"^^ ; "Lomax1.scale"^^ ; "\\lambda"^^ ; "\\lambda > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "Negative Binomial 6" ; "Negative Binomial 6"^^ ; "NegativeBinomial6"^^ ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a owl:ObjectProperty ; rdfs:label "distribution has SF" . a ; rdfs:label "relationship between Frechet 2 and Frechet 1 whereby m=0" ; ; ; "m=0"^^ ; "Frechet2(\\alpha,\\sigma,m) \\rightarrow Frechet1(\\alpha,\\sigma)"^^ ; "\\url{http://www.mathwave.com/help/easyfit/html/analyses/distributions/frechet.html}"^^ . a ; rdfs:label "Double Exponential 1"^^ , "Laplace 1" ; "Laplace 1"^^ ; "Laplace1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "HF of Exponential 1" ; "HF of Exponential 1"^^ ; "\\lambda"^^ ; "lambda"^^ . a ; rdfs:label "scale of Erlang-1" ; "scale of Erlang-1"^^ ; "Erlang1.scale"^^ ; "b"^^ ; "b>0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "relationship between Normal 1 and Logit Normal 1 whereby \\\\text{ If } Y \\\\text{ is a random variable with a normal distribution, and } P \\\\text{ is the logistic function, then } \\\\\\\\ X = PY \\\\text{ has a logit-normal distribution}" ; ; ; "\\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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Logit-normal_distribution}"^^ . a ; rdfs:label "Normal-inverse-gamma 1" , "Gaussian- inverse-gamma"^^ ; "Normal-inverse-gamma 1"^^ ; "NormalInverseGamma1"^^ ; ; "x"^^ ; "x \\in (-\\infty,+\\infty), \\sigma^2 \\in (0,+\\infty)"^^ ; , , , . a ; rdfs:label "variance of Normal 3" ; "variance of Normal 3"^^ ; "1/\\tau"^^ . a ; rdfs:label "PDF of Inverse-Gamma 1" ; "PDF of Inverse-Gamma 1"^^ ; "\\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{-\\alpha - 1} \\exp \\left(\\frac{-\\beta}{x}\\right)"^^ ; "beta^alpha/gamma(alpha) * x^(-alpha-1) * exp(-beta/x)"^^ . a ; rdfs:label "scale parameter of Rice-1" ; "scale parameter of Rice-1"^^ ; "Rice1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Zipf 1" ; "mean of Zipf 1"^^ ; "H_{n,\\alpha-1}/ H_{n,\\alpha}"^^ . a ; rdfs:label "relationship between Generalized Gamma 3 and Generalized Gamma 1 whereby a=1/\\\\mu, d=\\\\beta r, p=\\\\beta" ; ; ; "a=1/\\mu, d=\\beta r, p=\\beta"^^ ; "GeneralizedGamma3(r,\\mu,\\beta) \\rightarrow GeneralizedGamma1(a,d,p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Trapezoidal 1" ; "PDF of Trapezoidal 1"^^ ; """\\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 \\\\"""^^ ; """PMF1 = 2/(d+c-a-b)*(x-a)/(b-a) PMF2 = 2/(d+c-a-b) PMF3 = 2/(d+c-a-b)*(d-x)/(d-c) """^^ . a owl:Class ; rdfs:label "Transformation, Limiting" ; rdfs:subClassOf . a ; rdfs:label "shape of Zeta-1" ; "shape of Zeta-1"^^ ; "Zeta1.shape"^^ ; "\\alpha"^^ ; "\\alpha \\in R, \\alpha > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "relationship between Hypergeometric 1 and Binomial 1 whereby p = K/N, n=n, N \\\\rightarrow \\\\infty" ; ; ; "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}"""^^ . a ; rdfs:label "ExGaussian"^^ , "Exponentially modified Gaussian 1" ; "Exponentially modified Gaussian 1"^^ ; "ExponentiallyModifiedGaussian1"^^ ; ; ; ; ; "x"^^ ; "x \\in R"^^ ; , , . a ; rdfs:label "scale matrix of Inverse-Wishart-1" ; "scale matrix of Inverse-Wishart-1"^^ ; "InverseWishart1.scaleMatrix"^^ ; "\\Psi"^^ ; "\\Psi > 0, \\text{positive-definite matrix}"^^ ; "scale matrix"^^ ; "scaleMatrix"^^ . a ; rdfs:label "Pareto Type II" ; "Pareto Type II"^^ ; "ParetoTypeII1"^^ ; ; ; "x"^^ ; "x \\in [\\mu, +\\infty)"^^ ; , , . a ; rdfs:label "variance of Lomax 1" ; "variance of Lomax 1"^^ ; """{{\\lambda^2 \\alpha} \\over {(\\alpha-1)^2(\\alpha-2)}} \\text{ for } \\alpha > 2;\\\\ \\infty \\text{ for } 1 < \\alpha \\le 2; \\\\ \\text{Otherwise undefined}"""^^ . a ; rdfs:label "mean of Poisson 2" ; "mean of Poisson 2"^^ ; "\\exp(\\alpha)"^^ . a owl:ObjectProperty ; rdfs:label "distribution has HF" . a ; rdfs:label "mean of Inverse-Gaussian-1" ; "mean of Inverse-Gaussian-1"^^ ; "InverseGaussian1.mean"^^ ; "\\mu"^^ ; "\\mu > 0"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "CDF of Exponential 1" ; "CDF of Exponential 1"^^ ; "1 - \\exp(-\\lambda x)"^^ ; "1 - exp(-lambda*x)"^^ . a ; rdfs:label "shape of Erlang-1" ; "shape of Erlang-1"^^ ; "Erlang1.shape"^^ ; "c"^^ ; "c>0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "mean of Normal-3" ; "mean of Normal-3"^^ ; "Normal3.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "PDF of Laplace 1" ; "PDF of Laplace 1"^^ ; "\\frac{1}{2\\,b} \\exp \\left(-\\frac{|x-\\mu|}b \\right)"^^ ; "1/(2*b) * exp(- abs(x-mu)/b )"^^ . a ; rdfs:label "CDF of Zipf 1" ; "CDF of Zipf 1"^^ ; """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) }"""^^ . a ; rdfs:label "scale of Logistic-1" ; "scale of Logistic-1"^^ ; "Logistic1.scale"^^ ; "s"^^ ; "s > 0, s \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Inverse-Gamma 1" ; "CDF of Inverse-Gamma 1"^^ ; "\\frac{\\Gamma(\\alpha, \\beta/x)}{\\Gamma(\\alpha)}"^^ ; "Igamma(alpha, beta/x, lower=F) / gamma(alpha)"^^ . a ; rdfs:label "degrees of freedom of Inverse-Wishart-1" ; "degrees of freedom of Inverse-Wishart-1"^^ ; "InverseWishart1.degreesOfFreedom"^^ ; "\\nu"^^ ; "\\nu > p-1, \\nu \\in R"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a owl:Class ; rdfs:label "Limiting, Special case" ; rdfs:subClassOf . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Exponentially modified Gaussian 1" ; "mean of Exponentially modified Gaussian 1"^^ ; "\\mu + 1/\\lambda"^^ . a ; rdfs:label "relationship between Logistic 1 and Logistic 2 whereby \\\\tau=1/s" ; ; ; "\\tau=1/s"^^ ; "Logistic1(\\mu,s) \\rightarrow Logistic2(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Pareto Type II" ; "PDF of Pareto Type II"^^ ; "{\\alpha \\over \\lambda} \\left[{1+ {x-\\mu \\over \\lambda}}\\right]^{-(\\alpha+1)}"^^ ; "alpha/lambda* (1+ (x-mu)/lambda)^(-(alpha+1))"^^ . a ; rdfs:label "location of Von-Mises-1" ; "location of Von-Mises-1"^^ ; "VonMises1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "CDF of Generalized Poisson 3" ; "CDF of Generalized Poisson 3"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "location of Logistic-1" ; "location of Logistic-1"^^ ; "Logistic1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "PDF of F 1" ; "PDF of F 1"^^ ; "\\frac{\\Gamma(\\frac{n_1 + n_2}{2}) (\\frac{n_1}{n_2})^{n_1/2} x^{n_1/2-1}}{\\Gamma(\\frac{n_1}{2})\\Gamma(\\frac{n_2}{2})\\big[\\frac{n_1}{n_2}x+1\\big]^{(n_1+n_2)/2}}"^^ ; "gamma((n1 + n2)/2)*(n1/n2)^(n1/2)*x^(n1/2-1)/(gamma(n1/2)*gamma(n2/2)*(n1/n2*x+1)^((n1+n2)/2))"^^ . a ; rdfs:label "mean of Inverse-Gamma 1" ; "mean of Inverse-Gamma 1"^^ ; "\\frac{\\beta}{\\alpha-1} \\text{ for } \\alpha > 1"^^ . a ; rdfs:label "CDF of Laplace 1" ; "CDF of Laplace 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}"""^^ ; """1/2 * exp( (x-mu)/b ) # for x < mu 1- 1/2 * exp( -(x-mu)/b ) # x >= mu"""^^ . a ; rdfs:label "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)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label """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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "precision of Normal-3" ; "precision of Normal-3"^^ ; "Normal3.precision"^^ ; "\\tau"^^ ; "\\tau>0"^^ ; "precision"^^ ; "precision"^^ . a ; rdfs:label "mean of Noncentral chi-squared 1" ; "mean of Noncentral chi-squared 1"^^ ; "\\delta + n"^^ . a ; rdfs:label "CDF of Exponentially modified Gaussian 1" ; "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)))) }"""^^ . a ; rdfs:label "Normal"^^ , "Gaussian"^^ , "Normal 1" ; "Normal 1"^^ ; "Normal1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in R"^^ ; , . a owl:Class ; rdfs:label "Special case, Limiting" ; rdfs:subClassOf . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "relationship between Erlang 1 and Exponential 2 whereby c=1, b=\\\\beta" ; ; ; "c=1, b=\\beta"^^ ; "Erlang1(b,c) \\rightarrow Exponential2(\\beta)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "PMF of Negative Binomial 6" ; "PMF of Negative Binomial 6"^^ ; "\\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"^^ . a ; rdfs:label "CDF of Pareto Type II" ; "CDF of Pareto Type II"^^ ; "1- \\left[{1+ {x-\\mu \\over \\lambda}}\\right]^{-\\alpha}"^^ ; "1- (1+ (x-mu)/lambda)^(-alpha)"^^ . a ; rdfs:label "mode of Arcsine 2" ; "mode of Arcsine 2"^^ ; "x \\in \\{a,b\\}"^^ . a ; rdfs:label "variance of Logistic 1" ; "variance of Logistic 1"^^ ; "\\frac{s^2 \\pi^2}{3}"^^ . a ; rdfs:label "variance of Skew Normal" ; "variance of Skew Normal"^^ ; "\\sigma^2\\left(1 - \\frac{2\\delta^2}{\\pi}\\right) \\text{ where } \\delta = \\frac{\\alpha}{\\sqrt{1+\\alpha^2}}"^^ . a ; rdfs:label "relationship between Standard Uniform 1 and Pareto Type I whereby x_m X^{-1/\\\\alpha}" ; ; ; "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}"""^^ . a ; rdfs:label "mode of Inverse-Gamma 1" ; "mode of Inverse-Gamma 1"^^ ; "\\frac{\\beta}{\\alpha + 1}"^^ . a ; rdfs:label "mean of Laplace 1" ; "mean of Laplace 1"^^ ; "\\mu"^^ . a ; rdfs:label "F 1" , "Fisher-Snedecor"^^ ; "F 1"^^ ; "F1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; , . a ; rdfs:label "relationship between Zero-inflated Poisson 1 and Poisson 1 whereby \\\\pi=0" ; ; ; "\\pi=0"^^ ; "ZeroInflatedPoisson1(\\lambda,\\pi) \\rightarrow Poisson1(\\lambda)"^^ ; "\\cite{Troconiz:2009fv}"^^ . a ; rdfs:label "CDF of Frechet 1" ; "CDF of Frechet 1"^^ ; "\\exp\\Big(-\\Big(\\frac{\\sigma}{x}\\Big)^\\alpha\\Big)"^^ ; "exp(-(sigma/x)^alpha)"^^ . a ; rdfs:label "Chi-square"^^ , "Chi-squared 1" , "Chi square"^^ , "Chisquare"^^ ; "Chi-squared 1"^^ ; "ChiSquared1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a ; rdfs:label "relationship between Conway-Maxwell-Poisson 1 and Binomial 1 whereby \\\\text{For } \\\\nu=\\\\infty \\\\text{ the distribution of the sum is binomial with parameters } \\\\\\\\ n \\\\text{ and } \\\\lambda/1+\\\\lambda" ; ; ; "\\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) "^^ ; "\\cite{shmueli2005useful}"^^ . a ; rdfs:label "shape of Frechet-2" ; "shape of Frechet-2"^^ ; "Frechet2.shape"^^ ; "\\alpha"^^ ; "\\alpha \\in (0, \\infty)"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "CDF of Skew Normal" ; "CDF of Skew Normal"^^ ; "\\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)"^^ . a ; rdfs:label "relationship between Zipf 1 and Zeta 1 whereby \\\\text{The limiting distribution of a } Zipf1\\\\alpha, n \\\\text{ random variable as }\\\\\\\\ n\\\\rightarrow \\\\infty \\\\text{ is the } Zeta1\\\\alpha \\\\text{ distribution.}" ; ; ; "\\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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ZipfZeta.pdf}"^^ . a ; rdfs:label "mean of Arcsine 2" ; "mean of Arcsine 2"^^ ; "(a+b)/2"^^ . a ; rdfs:label "shape parameter of Beta-Negative-Binomial1" ; "shape parameter of Beta-Negative-Binomial1"^^ ; "BetaNegativeBinomial1.shape2"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "variance of Logistic 2" ; "variance of Logistic 2"^^ ; "\\frac{\\pi^2}{3\\tau^2 }"^^ . a ; rdfs:label "relationship between Student's t-distribution 3 and Student's t-distribution 1 whereby \\\\mu=0, \\\\sigma=1" ; ; ; "\\mu=0, \\sigma=1"^^ ; "StudentT3(\\nu,\\mu,\\sigma) \\rightarrow StudentT1(\\nu)"^^ ; "ProbOnto spec"^^ . a owl:DatatypeProperty ; rdfs:label "has R code expression" . a owl:Class ; rdfs:label "cumulative density function" ; rdfs:subClassOf . a ; rdfs:label "Normal"^^ , "Gaussian"^^ , "Normal 2" ; "Normal 2"^^ ; "Normal2"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in R"^^ ; , . a ; rdfs:label "relationship between Chi-squared 1 and F 1 whereby \\\\text{If } X_1 \\\\sim ChiSquared1n_1, X_2 \\\\sim ChiSquared1n_2 \\\\text{ are independent random variables }\\\\\\\\ \\\\Rightarrow \\\\frac{X_1/n_1}{X_2/n_2} \\\\sim F1n_1,n_2" ; ; ; "\\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)"^^ ; "ChiSquared1(n) \\rightarrow F1(n_1,n_2)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ChisquareF.pdf}"""^^ . a ; rdfs:label "PDF of Triangular 1" ; "PDF of Triangular 1"^^ ; """\\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}"""^^ ; """2*(x-a) / ((b-a)*(c-a)) for a <= x <= c \\\\ 2*(b-x) / ((b-a)*(b-c)) for c <= x <= b """^^ . a ; rdfs:label "relationship between Negative Binomial 1 and Negative Binomial 3 whereby \\\\phi=r, \\\\mu=r1-p/p" ; ; ; "\\phi=r, \\mu=r(1-p)/p"^^ ; "NegativeBinomial1(r, p) \\rightarrow NegativeBinomial3(\\mu, \\phi)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Student's t-distribution 1" ; "PDF of Student's t-distribution 1"^^ ; "\\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}}"^^ ; "gamma((nu+1)/2)/(sqrt(nu*pi)*gamma(nu/2))*(1+x^2/nu)^(-(nu+1)/2)"^^ . a ; rdfs:label "number of successes of Beta-Negative-Binomial1" ; "number of successes of Beta-Negative-Binomial1"^^ ; "BetaNegativeBinomial1.numberOfSuccesses"^^ ; "n"^^ ; "n \\in N"^^ ; "number of successes"^^ ; "numberOfSuccesses"^^ . a ; rdfs:label "location of Standard-Two-Sided-Power-1" ; "location of Standard-Two-Sided-Power-1"^^ ; "StandardTwoSidedPower1.location"^^ ; "m"^^ ; "a < m < b"^^ ; "location"^^ ; "location"^^ . a owl:ObjectProperty ; rdfs:label "distribution has variance" . a ; rdfs:label "shape of Gamma-1" ; "shape of Gamma-1"^^ ; "Gamma1.shape"^^ ; "k"^^ ; "k > 0"^^ ; "shape"^^ ; "shape"^^ . a owl:Class ; rdfs:label "relationship between distributions" . a owl:Class ; rdfs:label "number" ; rdfs:subClassOf . a ; rdfs:label "PDF of Log-Logistic 1" ; "PDF of Log-Logistic 1"^^ ; "\\frac{(\\beta/\\alpha)(x/\\alpha)^{\\beta-1}}{(1+(x/\\alpha)^\\beta)^2}"^^ ; "(beta/alpha)*(x/alpha)^(beta-1) / (1+(x/alpha)^beta)^2"^^ . a ; rdfs:label "Rayleigh 1" ; "Rayleigh 1"^^ ; "Rayleigh1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "relationship between Triangular 1 and Standard Triangular 1 whereby StandardTriangular1 \\\\text{ distribution is a special case of the } Triangular1a,b,c \\\\text{ distribution } \\\\\\\\ \\\\text{ when } a = -1, b = 1, c = 0." ; ; ; "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"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TriangularStandardtriangular.pdf}"^^ . a ; rdfs:label "mean of Skew Normal" ; "mean of Skew Normal"^^ ; "\\mu + \\sigma\\delta\\sqrt{\\frac{2}{\\pi}} \\text{ where } \\delta = \\frac{\\alpha}{\\sqrt{1+\\alpha^2}}"^^ . a ; rdfs:label "CDF of Arcsine 2" ; "CDF of Arcsine 2"^^ ; "\\frac{2}{\\pi}\\arcsin\\left(\\sqrt \\frac{x-a}{b-a} \\right)"^^ ; "2/pi*asin(sqrt((x-a)/(b-a)))"^^ . a ; rdfs:label "location of Logistic-2" ; "location of Logistic-2"^^ ; "Logistic2.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "scale of Laplace-1" ; "scale of Laplace-1"^^ ; "Laplace1.scale"^^ ; "b"^^ ; "b > 0, b \\in R"^^ ; "scale"^^ ; "scale"^^ . a owl:Class ; rdfs:label "probability mass function" ; rdfs:subClassOf . a owl:DatatypeProperty ; rdfs:label "has abbreviation" . a ; rdfs:label "relationship between Negative Binomial 2 and Negative Binomial 1 whereby r=1/\\\\tau, p = 1/1+\\\\tau \\\\lambda" ; ; ; "r=1/\\tau, p = 1/(1+\\tau \\lambda)"^^ ; "NegativeBinomial2(\\lambda, \\tau) \\rightarrow NegativeBinomial1(r, p) "^^ ; "ProbOnto spec"^^ . a ; rdfs:label "relationship between Normal 1 and Maxwell Boltzmann 1 whereby \\\\text{If } U_1, U_2, U_3 \\\\text{ are independent normal variables with mean} 0 \\\\text{ and standard deviation } \\\\sigma > 0 \\\\text{ then } X = \\\\sqrt{U_1^2 +U_3^2 +U_3^2} \\\\text{ has the Maxwell distribution with scale parameter } \\\\sigma" ; ; ; "\\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)"^^ ; "\\url{http://www.math.uah.edu/stat/special/Maxwell.html}"^^ . a ; rdfs:label "CDF of Student's t-distribution 1" ; "CDF of Student's t-distribution 1"^^ ; "\\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)}"^^ ; "1/2+x*gamma((nu+1)/2)*hypergeo(1/2,(nu+1)/2,3/2,-x^2/nu)/( sqrt(pi*nu) *gamma(nu/2))"^^ . a ; rdfs:label "CDF of Triangular 1" ; "CDF of Triangular 1"^^ ; """\\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}"""^^ ; """(x-a)^2 / ((b-a)*(c-a)) for a <= x <= c \\\\ 1 - (b-x)^2 / ((b-a)*(b-c)) for c <= x <= b """^^ . a ; rdfs:label "relationship between Student's t-distribution 2 and Student's t-distribution 1 whereby \\\\mu=0, \\\\tau=1" ; ; ; "\\mu=0, \\tau=1"^^ ; "StudentT2(\\mu,\\tau,k) \\rightarrow StudentT1(\\nu)"^^ . a ; rdfs:label "HF of Geometric 1" ; "HF of Geometric 1"^^ ; "p"^^ ; "p"^^ . a owl:ObjectProperty ; rdfs:label "distribution has mode" . a ; rdfs:label "Log-Logistic 1" , "Fisk"^^ ; "Log-Logistic 1"^^ ; "LogLogistic1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; , . a ; rdfs:label "variance of Gamma 1" ; "variance of Gamma 1"^^ ; "k \\theta^2"^^ . a ; rdfs:label "variance of Standard Two-Sided Power 1" ; "variance of Standard Two-Sided Power 1"^^ ; "\\frac{n-2(n-1)m(1-m)}{(n+2)(n+1)^2}"^^ . a owl:Class ; rdfs:label "mathematical object" . a owl:ObjectProperty ; rdfs:label "distribution has parameter" . a ; rdfs:label "PDF of Scaled Inverse Chi-Square" ; "PDF of Scaled Inverse Chi-Square"^^ ; "\\frac{(\\nu/2)^{\\nu/2}}{\\Gamma(\\nu/2)}\\sigma^\\nu x^{-(\\nu/2+1)} \\exp\\Big(-\\frac{1}{2}\\nu \\sigma^2 \\frac{1}{x}\\Big)"^^ ; "(nu/2)^(nu/2)/gamma(nu/2)*sigma^nu*x^(-(nu/2+1))*exp(-1/2*nu*sigma^2*1/x)"^^ . a ; rdfs:label "PDF of Rayleigh 1" ; "PDF of Rayleigh 1"^^ ; "\\frac{x}{\\sigma^2} e^{-x^2/(2\\sigma^2)}"^^ ; "x/sigma^2 * exp(-x^2/(2*sigma^2))"^^ . a owl:Class ; rdfs:label "integer" ; rdfs:subClassOf . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Kumaraswamy_distribution}"^^ . a ; rdfs:label "Skew Normal" ; "Skew Normal"^^ ; "SkewNormal1"^^ ; ; ; ; ; "x"^^ ; "x \\in R"^^ ; , , . a ; rdfs:label """relationship between Log-Normal 3 and Log-Normal 6 whereby m = m, \\\\sigma_g=\\\\exp\\\\sigma""" ; ; ; """m = m, \\sigma_g=\\exp(\\sigma)"""^^ ; "LogNormal3(m,\\sigma) \\rightarrow LogNormal6(m,\\sigma_g)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Normal 2" ; "CDF of Normal 2"^^ ; "\\frac12\\left[1 + \\text{erf}\\left( \\frac{x-\\mu}{\\sqrt{v}\\sqrt{2}}\\right)\\right]"^^ ; "1/2 * (1 + erf((x-mu)/(sqrt(var)*sqrt(2))))"^^ . a ; rdfs:label "relationship between Beta 1 and Standard Uniform 1 whereby \\\\alpha=1, \\\\beta = 1" ; ; ; "\\alpha=1, \\beta = 1"^^ ; "Beta1(\\alpha,\\beta) \\rightarrow StandardUniform1"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "inverse scale of Logistic-2" ; "inverse scale of Logistic-2"^^ ; "Logistic2.inverseScale"^^ ; "\\tau"^^ ; "\\tau > 0, \\tau \\in R"^^ ; "inverse scale"^^ ; "inverseScale"^^ . a owl:Class ; rdfs:label "probability density function" ; rdfs:subClassOf . a ; rdfs:label "CDF of Standard Two-Sided Power 1" ; "CDF of Standard Two-Sided Power 1"^^ ; """\\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}"""^^ ; """CDF1 = function(x,m,n) { x^n * m^(1-n) } CDF2 = function(x,m,n) { 1-(1-m)*((1-x)/(1-m))^n }"""^^ . a ; rdfs:label "mean of Beta Negative Binomial1" ; "mean of Beta Negative Binomial1"^^ ; """\\begin{cases} \\frac{n\\beta}{\\alpha-1} & \\text{if}\\ \\alpha>1 \\\\ \\infty & \\text{otherwise}\\ \\end{cases}"""^^ . a ; rdfs:label "mode of Wigner Semicircle 1" ; "mode of Wigner Semicircle 1"^^ ; "0"^^ . a ; rdfs:label "mean of Standard Two-Sided Power 1" ; "mean of Standard Two-Sided Power 1"^^ ; "\\frac{(n-1)m+1}{n+1}"^^ . a ; rdfs:label "upper bound of Truncated-Normal-1" ; "upper bound of Truncated-Normal-1"^^ ; "TruncatedNormal1.upperBound"^^ ; "b"^^ ; "b \\in R, b > a"^^ ; "upper bound"^^ ; "upperBound"^^ . a ; rdfs:label "mode of Gamma 1" ; "mode of Gamma 1"^^ ; "(k \\,-\\, 1)\\theta \\text{ for } k \\;{\\geq}\\; 1"^^ . a ; rdfs:label "shape of Lomax-1" ; "shape of Lomax-1"^^ ; "Lomax1.shape"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label """relationship between Log-Normal 1 and Log-Normal 6 whereby m=\\\\exp\\\\mu, \\\\sigma_g=\\\\exp\\\\sigma""" ; ; ; """m=\\exp(\\mu), \\sigma_g=\\exp(\\sigma)"""^^ ; "LogNormal1(\\mu,\\sigma) \\rightarrow LogNormal6(m,\\sigma_g)"^^ ; "ProbOnto spec"^^ . a owl:ObjectProperty ; rdfs:label "distribution has median" . a ; rdfs:label "event probabilities of Multinomial-1" ; "event probabilities of Multinomial-1"^^ ; "Multinomial1.probabilityOfSuccess"^^ ; "p_1, \\ldots, p_k"^^ ; "p_1, \\ldots, p_k, \\Sigma p_i = 1"^^ ; "event probabilities"^^ ; "probabilityOfSuccess"^^ . a ; rdfs:label "CDF of Rayleigh 1" ; "CDF of Rayleigh 1"^^ ; "1 - e^{-x^2/(2\\sigma^2)}"^^ ; "1 - exp(-x^2/(2*sigma^2))"^^ . a owl:Class ; rdfs:label "Reparameterisation" ; rdfs:subClassOf . a ; rdfs:label "relationship between Zipf 1 and Uniform Discrete 1 whereby \\\\alpha = 0, a = 1, b = n" ; ; ; "\\alpha = 0, a = 1, b = n"^^ ; "Zipf1(\\alpha,n) \\rightarrow UniformDiscrete1(a,b)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ZipfDiscreteuniform.pdf}"^^ . a owl:Class ; rdfs:label "natural number" ; rdfs:subClassOf . a ; rdfs:label "median of Arcsine 2" ; "median of Arcsine 2"^^ ; "(a+b)/2"^^ . a ; rdfs:label "PDF of Skew Normal" ; "PDF of Skew Normal"^^ ; "\\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]"^^ ; "1/(sigma*sqrt(2*pi)) * exp(-1/2*((x-mu)/sigma)^2) * (1+erf(alpha*((x-mu)/(sigma*sqrt(2)))))"^^ . a owl:Class ; rdfs:label "parameterised probability distribution" ; rdfs:subClassOf . a ; rdfs:label "relationship between NoncentralT and Student's t-distribution 1 whereby \\\\delta=0" ; ; ; "\\delta=0"^^ ; "NoncentralT1(\\delta,n) \\rightarrow StudentT1(n)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NoncentraltT.pdf}"^^ . a ; rdfs:label "PDF of Normal 2" ; "PDF of Normal 2"^^ ; "\\frac{1}{\\sqrt{v} \\sqrt{2 \\pi}}e^{-\\frac{(x-\\mu)^2}{2v}}"^^ ; "1/(sqrt(var)*sqrt(2*pi))*exp(-(x-mu)^2/(2*var))"^^ . a ; rdfs:label "PDF of Standard Two-Sided Power 1" ; "PDF of Standard Two-Sided Power 1"^^ ; """\\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}"""^^ ; """PDF1 = function(x,m,n) { n*(x/m)^(n-1) } PDF2 = function(x,m,n) { n*((1-x)/(1-m))^(n-1) }"""^^ . a ; rdfs:label "PDF of Johnson SU 1" ; "PDF of Johnson SU 1"^^ ; "\\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))"^^ . a ; rdfs:label "shape parameter of Beta-Negative-Binomial1" ; "shape parameter of Beta-Negative-Binomial1"^^ ; "BetaNegativeBinomial1.shape1"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "relationship between Log-Normal 2 and Log-Normal 1 whereby \\\\mu = \\\\mu, \\\\sigma = \\\\sqrt{v}" ; ; ; "\\mu = \\mu, \\sigma = \\sqrt{v}"^^ ; "LogNormal2(\\mu,v) \\rightarrow LogNormal1(\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "lower bound of Truncated-Normal-1" ; "lower bound of Truncated-Normal-1"^^ ; "TruncatedNormal1.lowerBound"^^ ; "a"^^ ; "a \\in R"^^ ; "lower bound"^^ ; "lowerBound"^^ . a ; rdfs:label "Epanechnikov 1" ; "Epanechnikov 1"^^ ; "Epanechnikov1"^^ ; ; ; ; ; "x"^^ ; "x \\in [-1,1]"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Triangular 1" ; "Triangular 1"^^ ; "Triangular1"^^ ; ; ; ; ; ; "x"^^ ; "a \\leq x \\leq b"^^ ; , , . a ; rdfs:label "t-distribution"^^ , "Student's t-distribution 1" ; "Student's t-distribution 1"^^ ; "StudentT1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; . a owl:ObjectProperty ; rdfs:label "distribution has mean" . a ; rdfs:label "mean of Rayleigh 1" ; "mean of Rayleigh 1"^^ ; "\\sigma \\sqrt{\\frac{\\pi}{2}}"^^ . a ; rdfs:label "median of Gamma 1" ; "median of Gamma 1"^^ ; "\\text{No simple closed form}"^^ . a ; rdfs:label "CDF of Log-Logistic 1" ; "CDF of Log-Logistic 1"^^ ; "\\frac{1}{1+(x/\\alpha)^{-\\beta}}"^^ ; "1 / (1+(x/alpha)^(-beta))"^^ . a owl:Class ; rdfs:label "Special case" ; rdfs:subClassOf . a owl:Class ; rdfs:label "parametric probability distribution" ; rdfs:subClassOf . a owl:DatatypeProperty ; rdfs:label "parameter has definition expressed in plain text" . a ; rdfs:label "PDF of Weibull 1" ; "PDF of Weibull 1"^^ ; "\\frac{k}{\\lambda}\\left(\\frac{x}{\\lambda}\\right)^{k-1}e^{-(x/\\lambda)^{k}}"^^ ; "k/lambda * (x/lambda)^(k-1) * exp(-(x/lambda)^k)"^^ . a ; rdfs:label "mode of Laplace 1" ; "mode of Laplace 1"^^ ; "\\mu"^^ . a ; rdfs:label "mean of Wigner Semicircle 1" ; "mean of Wigner Semicircle 1"^^ ; "0"^^ . a ; rdfs:label "Minimax distribution"^^ , "Kumaraswamy 1" ; "Kumaraswamy 1"^^ ; "Kumaraswamy1"^^ ; ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,1]"^^ ; , . a ; rdfs:label "PDF of Log-Normal 6" ; "PDF of Log-Normal 6"^^ ; "\\frac{1}{x\\log(\\sigma_g)\\sqrt{2 \\pi}} \\exp\\Big[ \\frac{-[\\log(x/m)]^2}{2 \\log^2(\\sigma_g)}\\Big]"^^ ; "1/(x*log(sigma_g)*sqrt(2*pi))*exp(-(log(x/m))^2/(2*log(sigma_g)^2))"^^ . a ; rdfs:label "mode of Student's t-distribution 1" ; "mode of Student's t-distribution 1"^^ ; "0"^^ . a ; rdfs:label "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)"^^ ; "\\cite{LavielleBook:2014}"^^ . a ; rdfs:label "variance of Negative Binomial 1" ; "variance of Negative Binomial 1"^^ ; "\\frac{r(1-p)}{p^2}"^^ . a ; rdfs:label "mean of logx of Log-Normal-5" ; "mean of log(x) of Log-Normal-5"^^ ; "LogNormal5.meanLog"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean of log(x)"^^ ; "meanLog"^^ . a ; rdfs:label "shape of Log-Normal-3" ; "shape of Log-Normal-3"^^ ; "LogNormal3.stdevLog"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "shape"^^ ; "stdevLog"^^ . a ; rdfs:label "variance of Arcsine 1" ; "variance of Arcsine 1"^^ ; "\\frac{1}{8}"^^ . a ; rdfs:label "PDF of Uniform 1" ; "PDF of Uniform 1"^^ ; """\\begin{cases} \\frac{1}{b - a} & \\text{for } x \\in [a,b] \\\\ 0 & \\text{otherwise} \\end{cases}"""^^ ; "1/(b-a)"^^ . a ; rdfs:label "mean of Gamma 1" ; "mean of Gamma 1"^^ ; "k \\theta"^^ . a owl:Class ; rdfs:label "non numerical set" . a ; rdfs:label "mode of Scaled Inverse Chi-Square" ; "mode of Scaled Inverse Chi-Square"^^ ; "\\frac{\\nu \\sigma^2}{\\nu + 2}"^^ . a ; rdfs:label "CDF of Normal 1" ; "CDF of Normal 1"^^ ; "\\frac12\\left[1 + \\text{erf}\\left( \\frac{x-\\mu}{\\sigma\\sqrt{2}}\\right)\\right]"^^ ; "1/2 * (1 + erf((x-mu)/(sigma*sqrt(2))))"^^ . a ; rdfs:label "shape of Log-Logistic-2" ; "shape of Log-Logistic-2"^^ ; "LogLogistic2.shape"^^ ; "\\kappa"^^ ; "\\kappa > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "median of Rayleigh 1" ; "median of Rayleigh 1"^^ ; "\\sigma \\sqrt{\\log(4)}"^^ . a ; rdfs:label "variance of Standard Cauchy 1" ; "variance of Standard Cauchy 1"^^ ; "undefined"^^ . a ; rdfs:label "PMF of Weibull Discrete 1" ; "PMF of Weibull Discrete 1"^^ ; "(1-p)^{x^\\beta} - (1-p)^{(x+1)^\\beta}"^^ ; "(1-p)^(x^beta) - (1-p)^((x+1)^beta)"^^ . a ; rdfs:label "relationship between Binomial 1 and Binomial 2 whereby \\\\alpha = \\\\logp/1-p" ; ; ; "\\alpha = \\log(p/(1-p))"^^ ; "Binomial1(n,p) \\rightarrow Binomial2(n,\\alpha)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "median of Laplace 1" ; "median of Laplace 1"^^ ; "\\mu"^^ . a ; rdfs:label "Weibull 1" ; "Weibull 1"^^ ; "Weibull1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; , . a ; rdfs:label "relationship between Gamma 1 and Erlang 1 whereby k \\\\in N, k=c, \\\\theta=b" ; ; ; "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}"""^^ . a owl:DatatypeProperty ; rdfs:label "parameter has definition" . a owl:Class ; rdfs:label "probability distribution" ; rdfs:subClassOf . a ; rdfs:label "mode of Arcsine 1" ; "mode of Arcsine 1"^^ ; "x \\in \\{0,1\\}"^^ . a ; rdfs:label "variance of Student's t-distribution 1" ; "variance of Student's t-distribution 1"^^ ; """\\begin{cases} \\frac{\\nu}{\\nu - 2} & \\text{for }\\nu > 2 \\\\ \\infty & \\text{for } 1< \\nu \\leq 2 \\\\ undefined & \\text{else} \\end{cases}"""^^ . a ; rdfs:label "PDF of Wiener Diffusion Model 1" ; "PDF of Wiener Diffusion Model 1"^^ ; "\\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)"^^ . a ; rdfs:label "median of Wigner Semicircle 1" ; "median of Wigner Semicircle 1"^^ ; "0"^^ . a ; rdfs:label "lognormal"^^ , "Galton"^^ , "Log-Normal 6" ; "Log-Normal 6"^^ ; "LogNormal6"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "mode of Negative Binomial 1" ; "mode of Negative Binomial 1"^^ ; """\\lfloor \\frac{(1-p)(r-1)}{p} \\rfloor """^^ . a ; rdfs:label "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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/MuthExponential.pdf}"^^ . a ; rdfs:label "precision of Log-Normal-5" ; "precision of Log-Normal-5"^^ ; "LogNormal5.precision"^^ ; "\\tau"^^ ; "\\tau > 0"^^ ; "precision"^^ ; "precision"^^ . a ; rdfs:label "Rician"^^ , "Rice 1" ; "Rice 1"^^ ; "Rice1"^^ ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; , . a ; rdfs:label "Uniform 1" , "Rectangular Continuous"^^ ; "Uniform 1"^^ ; "Uniform1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [a,b]"^^ ; , . a ; rdfs:label "variance of Scaled Inverse Chi-Square" ; "variance of Scaled Inverse Chi-Square"^^ ; "\\frac{2\\nu^2 \\sigma^4}{(\\nu - 2)^2(\\nu-4)} \\text{ for } \\nu>4"^^ . a ; rdfs:label "CDF of Gamma 1" ; "CDF of Gamma 1"^^ ; "\\frac{1}{\\Gamma(k)} \\gamma\\left(k,\\, \\frac{x}{\\theta}\\right)"^^ ; "1/gamma(k) * Igamma(k,x/theta)"^^ . a owl:Class ; rdfs:label "finite set" ; rdfs:subClassOf . a ; rdfs:label "CDF of Weibull Discrete 1" ; "CDF of Weibull Discrete 1"^^ ; "1-(1-p)^{(x+1)^\\beta}"^^ ; "1-(1-p)^((x+1)^beta)"^^ . a ; rdfs:label "Wiener Diffusion Model 1" ; "Wiener Diffusion Model 1"^^ ; "WienerDiffusionModel1"^^ ; ; "x"^^ ; , , , . a ; rdfs:label "mode of Rayleigh 1" ; "mode of Rayleigh 1"^^ ; "\\sigma"^^ . a ; rdfs:label "PDF of Normal 1" ; "PDF of Normal 1"^^ ; "\\frac{1}{\\sigma \\sqrt{2 \\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}"^^ ; "1/(sigma*sqrt(2*pi))*exp(-(x-mu)^2/(2*sigma^2))"^^ . a ; rdfs:label "mode of Standard Cauchy 1" ; "mode of Standard Cauchy 1"^^ ; "0"^^ . a ; rdfs:label "scale of Log-Logistic-2" ; "scale of Log-Logistic-2"^^ ; "LogLogistic2.scale"^^ ; "\\lambda"^^ ; "\\lambda > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "CDF of Uniform Discrete 1" ; "CDF of Uniform Discrete 1"^^ ; "(k-a+1)/(b-a+1)"^^ ; "(k-a+1)/(b-a+1)"^^ . a ; rdfs:label "PDF of Wigner Semicircle 1" ; "PDF of Wigner Semicircle 1"^^ ; "\\frac2{\\pi R^2} \\sqrt{R^2-x^2}"^^ ; "2/(pi * R^2) * sqrt(R^2-x^2)"^^ . a ; rdfs:label "variance of Log-Logistic 2" ; "variance of Log-Logistic 2"^^ ; "\\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}"^^ . a ; rdfs:label "PDF of Rice 1" ; "PDF of Rice 1"^^ ; "\\frac{x}{\\sigma^2}\\exp\\left(\\frac{-(x^2+\\nu^2)}{2\\sigma^2}\\right)I_0\\left(\\frac{x\\nu}{\\sigma^2}\\right)"^^ ; "x/sigma^2*exp(-(x^2+nu^2)/2/sigma^2) * besselI(x*nu/sigma^2, 0, expon.scaled = FALSE)"^^ . a ; rdfs:label "location of Laplace-1" ; "location of Laplace-1"^^ ; "Laplace1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "mode of Log-Normal 5" ; "mode of Log-Normal 5"^^ ; "e^{\\mu - \\frac{1}{\\tau}}"^^ . a ; rdfs:label "relationship between Logistic 2 and Logistic 1 whereby s=1/\\\\tau" ; ; ; "s=1/\\tau"^^ ; "Logistic2(\\mu,\\tau) \\rightarrow Logistic1(\\mu,s)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Arcsine 2" ; "PDF of Arcsine 2"^^ ; "\\frac{1}{\\pi\\sqrt{(x-a)(b-x)}}"^^ ; "1/pi/sqrt((x-a)*(b-x))"^^ . a ; rdfs:label "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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PowerStandardpower.pdf}"^^ . a ; rdfs:label "relationship between Negative Binomial 4 and Negative Binomial 5 whereby \\\\alpha = r, \\\\beta = 1-p / p" ; ; ; "\\alpha = r, \\beta = (1-p) / p"^^ ; "NegativeBinomial4(r,p) \\rightarrow NegativeBinomial5(\\alpha, \\beta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "location of minimum of Frechet-2" ; "location of minimum of Frechet-2"^^ ; "Frechet2.locationOfMinimum"^^ ; "m"^^ ; "m \\in (-\\infty, \\infty)"^^ ; "location of minimum"^^ ; "locationOfMinimum"^^ . a ; rdfs:label """relationship between Normal 1 and Folded Normal 1 whereby \\\\text{Given a normally distributed random variable X with mean } \\\\mu \\\\text{ and variance } \\\\sigma^2, \\\\text{ the random}\\\\\\\\ \\\\text{variable } Y = |X| \\\\text{ has a folded normal distribution}""" ; ; ; """\\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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Folded_normal_distribution}"^^ . a ; rdfs:label "CDF of Scaled Inverse Chi-Square" ; "CDF of Scaled Inverse Chi-Square"^^ ; "\\Gamma\\Big(\\frac{\\nu}{2},\\frac{\\sigma^2\\nu}{2x}\\Big)\\Big/\\Gamma\\Big(\\frac{\\nu}{2}\\Big)"^^ ; "Igamma(nu/2,sigma^2*nu/2/x, lower=F) / gamma(nu/2) \\text{ with } Igamma(a, x, lower=F)"^^ . a ; rdfs:label "relationship between Normal 1 and Standard Normal 1 whereby \\\\mu = 0, \\\\sigma = 1" ; ; ; "\\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}"""^^ . a owl:Class ; rdfs:label "mathematical set" ; rdfs:subClassOf . a ; rdfs:label "PDF of Gamma 1" ; "PDF of Gamma 1"^^ ; "\\frac{1}{\\Gamma(k) \\theta^k} x^{k \\,-\\, 1} e^{-\\frac{x}{\\theta}}"^^ ; "1 / (gamma(k) * theta^k) * x^(k-1) * exp(-x/theta)"^^ . a ; rdfs:label "vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-2" ; "vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-2"^^ ; "MultivariateGaussianProcess2.inverseScale"^^ ; "w"^^ ; "w \\in R^K, K \\in N"^^ ; "vector of positive inverse scales"^^ ; "inverseScale"^^ . a owl:ObjectProperty ; rdfs:label "distribution has nth parameter" . a ; rdfs:label "SF of Geometric 1" ; "SF of Geometric 1"^^ ; "(1 - p)^k"^^ ; "(1 - p)^k"^^ . a ; rdfs:label "median of Normal 1" ; "median of Normal 1"^^ ; "\\mu"^^ . a ; rdfs:label "Wigner Semicircle 1" ; "Wigner Semicircle 1"^^ ; "WignerSemicircle1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-R,R)"^^ ; . a ; rdfs:label "median of Standard Cauchy 1" ; "median of Standard Cauchy 1"^^ ; "0"^^ . a ; rdfs:label "mean of Student's t-distribution 1" ; "mean of Student's t-distribution 1"^^ ; """\\begin{cases} 0 & \\text{for }\\nu > 1 \\\\ undefined & \\text{else} \\end{cases}"""^^ . a ; rdfs:label "variance of Rayleigh 1" ; "variance of Rayleigh 1"^^ ; "\\frac{4 - \\pi}{2} \\sigma^2"^^ . a ; rdfs:label "median of Log-Logistic 2" ; "median of Log-Logistic 2"^^ ; "1/\\lambda"^^ . a ; rdfs:label "relationship between Negative Binomial 3 and Negative Binomial 6 whereby \\\\eta = \\\\log\\\\mu" ; ; ; "\\eta = \\log(\\mu)"^^ ; "NegativeBinomial3(\\mu,\\phi) \\rightarrow NegativeBinomial6(\\eta,\\phi)"^^ ; "\\cite{stan-manual:2015b}"^^ . a ; rdfs:label "boundary separation of Wiener-Diffusion-Model-1" ; "boundary separation of Wiener-Diffusion-Model-1"^^ ; "WienerDiffusionModel1.boundSeparation"^^ ; "\\alpha"^^ ; "\\alpha \\in R^+"^^ ; "boundary separation"^^ ; "boundSeparation"^^ . a ; rdfs:label "CDF of Wigner Semicircle 1" ; "CDF of Wigner Semicircle 1"^^ ; "\\frac12 \\frac{x\\sqrt{R^2-x^2}}{\\pi R^2} + \\frac{\\arcsin(\\frac{x}{R})}{\\pi}"^^ ; "1/2 * (x*sqrt(R^2-x^2))/(pi * R^2) + asin(x/R)/pi"^^ . a ; rdfs:label "median of Student's t-distribution 1" ; "median of Student's t-distribution 1"^^ ; "0"^^ . a ; rdfs:label "variance of Laplace 1" ; "variance of Laplace 1"^^ ; "2 b^2"^^ . a ; rdfs:label """relationship between Uniform 1 and Standard Uniform 1 whereby a = 0, b = 1""" ; ; ; """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}"""^^ . a ; rdfs:label "Arcsine 2" ; "Arcsine 2"^^ ; "Arcsine2"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (a,b)"^^ ; , . a ; rdfs:label "variance of Log-Normal 5" ; "variance of Log-Normal 5"^^ ; "e^{2\\mu + \\frac{1}{\\tau}}[e^{\\frac{1}{\\tau}}-1]"^^ . a ; rdfs:label "relationship between Two-Sided Power 1 and Standard Two-Sided Power 1 whereby a=0, b=1" ; ; ; "a=0, b=1"^^ ; "TwoSidedPower1(a,b,m,n) \\rightarrow StandardTwoSidedPower1(m,n)"^^ ; "\\cite{van2002standard}"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "scale of Frechet-2" ; "scale of Frechet-2"^^ ; "Frechet2.scale"^^ ; "\\sigma"^^ ; "\\sigma \\in (0, \\infty)"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "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"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardpowerStandarduniform.pdf}"^^ . a ; rdfs:label "mean of Scaled Inverse Chi-Square" ; "mean of Scaled Inverse Chi-Square"^^ ; "\\frac{\\nu \\sigma^2}{\\nu - 2} \\text{ for } \\nu>2"^^ . a ; rdfs:label "Gamma 1" ; "Gamma 1"^^ ; "Gamma1"^^ ; ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "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)"^^ ; "\\url{http://azzalini.stat.unipd.it/SN/Intro/intro.html}"^^ . a ; rdfs:label "lower triangular matrix of Multivariate-Gaussian-Process-Distribution-2" ; "lower triangular matrix of Multivariate-Gaussian-Process-Distribution-2"^^ ; "MultivariateGaussianProcess2.choleskyFactor"^^ ; "L"^^ ; "L \\in R^{N \\times N}, \\text{ lower triangular and such that } LL^T \\text{ is positive kernel definite}, \\text{with } N \\in N"^^ ; "lower triangular matrix"^^ ; "choleskyFactor"^^ . a owl:Class ; rdfs:label "numerical set" . a ; rdfs:label "mean of Standard Cauchy 1" ; "mean of Standard Cauchy 1"^^ ; "undefined"^^ . a ; rdfs:label "mean of Normal 1" ; "mean of Normal 1"^^ ; "\\mu"^^ . a owl:DatatypeProperty ; rdfs:label "has short code name" . a ; rdfs:label "Weibull Discrete 1" ; "Weibull Discrete 1"^^ ; "WeibullDiscrete1"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,\\dots\\}"^^ ; , . a owl:ObjectProperty ; rdfs:label "distribution has relation of type to" . a ; rdfs:label "scale of Rayleigh-1" ; "scale of Rayleigh-1"^^ ; "Rayleigh1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "mode of Logit Normal 1" ; "mode of Logit Normal 1"^^ ; "\\text{ no analytical solution }"^^ . a ; rdfs:label "CDF of Gumbel 1" ; "CDF of Gumbel 1"^^ ; "e^{-e^{-(x-\\mu)/\\beta}}"^^ ; "exp(-exp(-(x-mu)/beta))"^^ . a ; rdfs:label "relationship between Chi-squared 1 and Exponential 1 whereby k=2 \\\\text{ and } \\\\lambda=1/2" ; ; ; "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}"""^^ . a ; rdfs:label "mean of Weibull 1" ; "mean of Weibull 1"^^ ; "\\lambda \\, \\Gamma(1+1/k)"^^ . a ; rdfs:label "SF of Standard Cauchy 1" ; "SF of Standard Cauchy 1"^^ ; "\\frac{\\pi - 2\\arctan(x)}{2\\pi}"^^ ; "(pi - 2*atan(x))/2/pi"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "population size of Hypergeometric-1" ; "population size of Hypergeometric-1"^^ ; "Hypergeometric1.populationSize"^^ ; "N"^^ ; "N \\in \\left\\{0,1,2,\\dots\\right\\}"^^ ; "population size"^^ ; "populationSize"^^ . a ; rdfs:label "CDF of Truncated Normal 1" ; "CDF of Truncated Normal 1"^^ ; "\\frac{\\Phi(\\frac{x-\\mu}{\\sigma})-\\Phi(\\frac{a-\\mu}{\\sigma})}{\\Phi(\\frac{b-\\mu}{\\sigma})-\\Phi(\\frac{a-\\mu}{\\sigma})}"^^ ; """( 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 }"""^^ . a ; rdfs:label "number of trials of Beta-binomial-1" ; "number of trials of Beta-binomial-1"^^ ; "BetaBinomial1.numberOfTrials"^^ ; "n"^^ ; "n \\in N, n > 0"^^ ; "number of trials"^^ ; "numberOfTrials"^^ . a ; rdfs:label "variance of Beta-binomial 1" ; "variance of Beta-binomial 1"^^ ; "\\frac{n\\alpha\\beta(\\alpha+\\beta+n)}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}"^^ . a ; rdfs:label "mode of Multivariate Normal 3" ; "mode of Multivariate Normal 3"^^ ; "\\mu"^^ . a ; rdfs:label "relationship between Negative Binomial 1 and Geometric 1 whereby n=1" ; ; ; "n=1"^^ ; "NegativeBinomial1(r,p) \\rightarrow Geometric1(p)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalGeometric.pdf}"""^^ . a ; rdfs:label "PMF of Negative Binomial 1" ; "PMF of Negative Binomial 1"^^ ; "\\binom {k+r-1}k p^r (1-p)^k"^^ ; "choose(k+r-1,k) * p^r * (1-p)^k"^^ . a ; rdfs:label "scale of Student-s-t-distribution-2" ; "scale of Student-s-t-distribution-2"^^ ; "StudentT2.scale"^^ ; "\\tau"^^ ; "\\tau > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "mode of Uniform 1" ; "mode of Uniform 1"^^ ; "\\text{any value in }[a,b]"^^ . a ; rdfs:label "NA of Makeham-1" ; "NA of Makeham-1"^^ ; "Makeham1.gamma"^^ ; "\\gamma"^^ ; "\\gamma > 0"^^ ; "gamma"^^ . a ; rdfs:label "relationship between Log-Normal 1 and Log-Normal 2 whereby \\\\mu = \\\\mu, v = \\\\sigma^2 " ; ; ; "\\mu = \\mu, v = \\sigma^2 "^^ ; "LogNormal1(\\mu,\\sigma) \\rightarrow LogNormal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Multivariate Normal 2" ; "variance of Multivariate Normal 2"^^ ; "T^{-1}"^^ . a ; rdfs:label "mode of Log-Normal 6" ; "mode of Log-Normal 6"^^ ; "m / e^{\\log^2(\\sigma_g)}"^^ . a ; rdfs:label "SF of Kumaraswamy 1" ; "SF of Kumaraswamy 1"^^ ; "(1-x^a)^b"^^ ; "-(1-x^a)^b"^^ . a ; rdfs:label "Noncentral chi-squared 1" ; "Noncentral chi-squared 1"^^ ; "NoncentralChiSquared1"^^ ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "Uniform Discrete 1" , "Rectangular Discrete 1"^^ ; "Uniform Discrete 1"^^ ; "UniformDiscrete1"^^ ; ; ; ; ; ; "k"^^ ; "k \\in \\{a,a+1,...,b-1,b\\}"^^ ; , . a ; rdfs:label "initial bias of Wiener-Diffusion-Model-1" ; "initial bias of Wiener-Diffusion-Model-1"^^ ; "WienerDiffusionModel1.initialBias"^^ ; "\\beta"^^ ; "\\beta \\in [0,1]"^^ ; "initial bias"^^ ; "initialBias"^^ . a ; rdfs:label "mean of Standard Logistic 1" ; "mean of Standard Logistic 1"^^ ; "0"^^ . a ; rdfs:label "CDF of Log-Normal 7" ; "CDF of Log-Normal 7"^^ ; "\\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]"^^ ; "1/2 + 1/2 * erf( (log(x)-log(mu_N/sqrt(sigma_N^2/mu_N^2 + 1))) / sqrt(2 * log(1+sigma_N^2/mu_N^2)) )"^^ . a ; rdfs:label "variance of Logit Normal 1" ; "variance of Logit Normal 1"^^ ; "\\text{ no analytical solution }"^^ . a ; rdfs:label "variance of Hypergeometric 1" ; "variance of Hypergeometric 1"^^ ; "n{K\\over N}{(N-K)\\over N}{N-n\\over N-1}"^^ . a ; rdfs:label "mean of Gumbel 1" ; "mean of Gumbel 1"^^ ; "\\mu + \\beta\\,\\gamma_E; \\text{ with is Euler constant } \\gamma_E"^^ . a ; rdfs:label "HF of Standard Cauchy 1" ; "HF of Standard Cauchy 1"^^ ; "\\frac2{(1+x^2)(\\pi - 2\\arctan(x))}"^^ ; "2/(1+x^2)/(pi - 2*atan(x))"^^ . a ; rdfs:label "SF of Weibull 1" ; "SF of Weibull 1"^^ ; "\\exp(-(x/\\lambda)^k)"^^ ; "exp(-(x/lambda)^k)"^^ . a ; rdfs:label """relationship between Normal 1 and Normal 3 whereby \\\\mu = \\\\mu, \\\\tau = 1 / \\\\sigma^2 """ ; ; ; """\\mu = \\mu, \\tau = 1 / \\sigma^2 """^^ ; "Normal1(\\mu,\\sigma) \\rightarrow Normal3(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Truncated Normal 1" ; "PDF of Truncated Normal 1"^^ ; "\\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 }"""^^ . a ; rdfs:label "relationship between Gamma 1 and Gamma 2 whereby r=k, \\\\mu = 1/ \\\\theta" ; ; ; "r=k, \\mu = 1/ \\theta"^^ ; "Gamma1(k,\\theta) \\rightarrow Gamma2(r,\\mu) "^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Beta-binomial 1" ; "mean of Beta-binomial 1"^^ ; "\\frac{n\\alpha}{\\alpha+\\beta}"^^ . a ; rdfs:label "CDF of Standard Logistic 1" ; "CDF of Standard Logistic 1"^^ ; "\\frac{e^{x}}{1+e^{x}}"^^ ; "exp(x)/(1+exp(x))"^^ . a ; rdfs:label "median of Uniform 1" ; "median of Uniform 1"^^ ; "\\tfrac{1}{2}(a+b)"^^ . a ; rdfs:label "Negative Binomial 1" ; "Negative Binomial 1"^^ ; "NegativeBinomial1"^^ ; ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\} \\text{ -- number of failures}"^^ ; , . a ; rdfs:label "rate of decay of Conway-Maxwell-Poisson-1" ; "rate of decay of Conway-Maxwell-Poisson-1"^^ ; "ConwayMaxwellPoisson1.rateOfDecay"^^ ; "\\nu"^^ ; "\\nu \\ge 0"^^ ; "rate of decay"^^ ; "rateOfDecay"^^ . a ; rdfs:label "degrees of freedom of Student-s-t-distribution-2" ; "degrees of freedom of Student-s-t-distribution-2"^^ ; "StudentT2.degreesOfFreedom"^^ ; "k"^^ ; "k \\ge 2"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "drift rate of Wiener-Diffusion-Model-1" ; "drift rate of Wiener-Diffusion-Model-1"^^ ; "WienerDiffusionModel1.driftRate"^^ ; "\\delta"^^ ; "\\delta \\in R"^^ ; "drift rate"^^ ; "driftRate"^^ . a ; rdfs:label "location of Multivariate-Normal-2" ; "location of Multivariate-Normal-2"^^ ; "MultivariateNormal2.mean"^^ ; "\\mu"^^ ; "\\mu \\in R^k"^^ ; "location"^^ ; "mean"^^ . a ; rdfs:label "NA of Makeham-1" ; "NA of Makeham-1"^^ ; "Makeham1.kappa"^^ ; "\\kappa"^^ ; "\\kappa > 0"^^ ; "kappa"^^ . a ; rdfs:label "median of Log-Normal 5" ; "median of Log-Normal 5"^^ ; "e^\\mu"^^ . a ; rdfs:label "PDF of Noncentral chi-squared 1" ; "PDF of Noncentral chi-squared 1"^^ ; "\\sum^\\infty_{k=0} \\frac{e^{-\\delta/2} (\\delta/2)^k}{k!} \\frac{e^{-x/2} x^{(n+2k)/2-1}}{2^{(n+2k)/2} \\Gamma[(n+2k)/2]}"^^ ; """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) }"""^^ . a ; rdfs:label "HF of Kumaraswamy 1" ; "HF of Kumaraswamy 1"^^ ; "\\frac{a \\,b \\,x^{a-1}}{ (1-x^a)}"^^ ; "a*b*x^(a-1) / (1-x^a)"^^ . a ; rdfs:label "median of Log-Normal 6" ; "median of Log-Normal 6"^^ ; "m"^^ . a ; rdfs:label "PMF of Uniform Discrete 1" ; "PMF of Uniform Discrete 1"^^ ; "1/(b-a+1)"^^ ; "1/(b-a+1)"^^ . a ; rdfs:label "number of trials of Hypergeometric-1" ; "number of trials of Hypergeometric-1"^^ ; "Hypergeometric1.numberOfTrials"^^ ; "n"^^ ; "n \\in \\left\\{0,1,2,\\dots,N\\right\\}"^^ ; "number of trials"^^ ; "numberOfTrials"^^ . a ; rdfs:label "relationship between Zero-Inflated Negative Binomial 1 and Poisson 1 whereby p0=0, \\\\tau \\\\rightarrow 0" ; ; ; "p0=0, \\tau \\rightarrow 0"^^ ; "ZeroInflatedNegativeBinomial1(\\lambda,\\tau,p0) \\rightarrow Poisson1(\\lambda)"^^ ; "\\cite{Troconiz:2009fv}"^^ . a ; rdfs:label "CDF of Standard Cauchy 1" ; "CDF of Standard Cauchy 1"^^ ; "\\frac{\\pi + 2\\arctan(x)}{2\\pi}"^^ ; "(pi + 2*atan(x))/2/pi"^^ . a ; rdfs:label "median of Gumbel 1" ; "median of Gumbel 1"^^ ; "\\mu - \\beta\\,\\ln(\\ln(2))"^^ . a ; rdfs:label "location of Logit-Normal-1" ; "location of Logit-Normal-1"^^ ; "LogitNormal1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "HF of Weibull Discrete 1" ; "HF of Weibull Discrete 1"^^ ; "1-(1-p)^{(x+1)^\\beta-x^\\beta}"^^ ; "1-(1-p)^((x+1)^beta-x^beta)"^^ . a ; rdfs:label "location of Multivariate-Normal-1" ; "location of Multivariate-Normal-1"^^ ; "MultivariateNormal1.mean"^^ ; "\\mu"^^ ; "\\mu \\in R^k"^^ ; "location"^^ ; "mean"^^ . a ; rdfs:label "right beta parameter of Beta-binomial-1" ; "right beta parameter of Beta-binomial-1"^^ ; "BetaBinomial1.beta"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "right beta parameter"^^ ; "beta"^^ . a ; rdfs:label "Truncated Normal 1" ; "Truncated Normal 1"^^ ; "TruncatedNormal1"^^ ; ; ; ; ; "x"^^ ; "x \\in [a,b]"^^ ; , , , . a ; rdfs:label "relationship between Log-Normal 2 and Log-Normal 3 whereby m = \\\\exp\\\\mu, \\\\sigma = \\\\sqrt{v}" ; ; ; "m = \\exp(\\mu), \\sigma = \\sqrt{v}"^^ ; "LogNormal2(\\mu,v) \\rightarrow LogNormal3(m,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "upper limit of Triangular-1" ; "upper limit of Triangular-1"^^ ; "Triangular1.upperLimit"^^ ; "b"^^ ; "b \\in R, a < b"^^ ; "upper limit"^^ ; "upperLimit"^^ . a ; rdfs:label "PDF of Log-Normal 7" ; "PDF of Log-Normal 7"^^ ; "\\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)"^^ ; """1 / (x * sqrt(2 * pi * log(1+sigma_N^2/mu_N^2 ) )) * exp( -( log(x) - log(mu_N/sqrt(1+sigma_N^2/mu_N^2)) )^2 / (2*log(1+sigma_N^2/mu_N^2)) )"""^^ . a ; rdfs:label "Logistic 2" ; "Logistic 2"^^ ; "Logistic2"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "mean of Uniform 1" ; "mean of Uniform 1"^^ ; "\\tfrac{1}{2}(a+b)"^^ . a ; rdfs:label "PDF of Power 1" ; "PDF of Power 1"^^ ; "\\frac{\\beta x^{\\beta-1}}{\\alpha^\\beta}"^^ ; "(beta * x^(beta-1))/alpha^beta"^^ . a ; rdfs:label "PDF of Standard Logistic 1" ; "PDF of Standard Logistic 1"^^ ; "\\frac{e^{x}}{\\left(1+e^{x}\\right)^2}"^^ ; "exp(x) / (1+exp(x))^2"^^ . a ; rdfs:label "mean of Negative Binomial 1" ; "mean of Negative Binomial 1"^^ ; "\\frac{r(1-p)}{p}"^^ . a ; rdfs:label "mean of Log-Normal 6" ; "mean of Log-Normal 6"^^ ; "m\\,e^{\\frac{1}{2} \\log^2(\\sigma_g)}"^^ . a ; rdfs:label "CDF of Kumaraswamy 1" ; "CDF of Kumaraswamy 1"^^ ; "1-(1-x^a)^b"^^ ; "1-(1-x^a)^b"^^ . a ; rdfs:label "relationship between Generalized Negative Binomial 1 and Inverse Binomial 1 whereby \\\\beta=2, \\\\theta=1-p" ; ; ; "\\beta=2, \\theta=1-p"^^ ; "GeneralizedNegativeBinomial1(\\theta,\\beta,m) \\rightarrow InverseBinomial1(k,p)"^^ ; "\\cite{yanagimoto1989inverse}"^^ . a ; rdfs:label "CDF of Noncentral chi-squared 1" ; "CDF of Noncentral chi-squared 1"^^ ; """e^{-\\delta/2} \\sum^\\infty_{k=0} \\frac{(\\delta/2)^k}{k!} Q(x;n+2k)\\\\ \\text{ with } Q(x;p) = \\frac{\\gamma(p/2,x/2)}{\\Gamma(p/2)}"""^^ ; """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) }"""^^ . a ; rdfs:label """relationship between Log-Normal 2 and Log-Normal 5 whereby \\\\mu = \\\\mu, \\\\tau = 1 / v""" ; ; ; """\\mu = \\mu, \\tau = 1 / v"""^^ ; "LogNormal2(\\mu,v) \\rightarrow LogNormal5(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Multivariate Normal 2" ; "mean of Multivariate Normal 2"^^ ; "\\mu"^^ . a ; rdfs:label "HF of Weibull 1" ; "HF of Weibull 1"^^ ; "\\frac{k}\\lambda \\Big(\\frac{x}\\lambda\\Big)^{k-1}"^^ ; "k/lambda *(x/lambda)^(k-1)"^^ . a ; rdfs:label "variance of Student's t-distribution 2" ; "variance of Student's t-distribution 2"^^ ; "\\frac{1}{\\tau} \\frac{k}{k-2} \\text{ for } k > 2"^^ . a ; rdfs:label "nondecision time of Wiener-Diffusion-Model-1" ; "nondecision time of Wiener-Diffusion-Model-1"^^ ; "WienerDiffusionModel1.nondecisionTime"^^ ; "\\tau"^^ ; "\\tau \\in R^+"^^ ; "nondecision time"^^ ; "nondecisionTime"^^ . a ; rdfs:label "number of successes of Hypergeometric-1" ; "number of successes of Hypergeometric-1"^^ ; "Hypergeometric1.numberOfSuccesses"^^ ; "K"^^ ; "K \\in \\left\\{0,1,2,\\dots,N\\right\\} "^^ ; "number of successes"^^ ; "numberOfSuccesses"^^ . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Log-logistic_distribution#Related_distributions}"^^ . a ; rdfs:label "covariance matrix of Multivariate-Normal-1" ; "covariance matrix of Multivariate-Normal-1"^^ ; "MultivariateNormal1.covarianceMatrix"^^ ; "\\Sigma"^^ ; "\\Sigma \\in R^{k\\times k}"^^ ; "covariance matrix"^^ ; "covarianceMatrix"^^ . a ; rdfs:label "mean of Folded Normal 1" ; "mean of Folded Normal 1"^^ ; " \\sigma \\sqrt{\\tfrac{2}{\\pi}} \\, e^{(-\\mu^2/2\\sigma^2)} + \\mu \\left(1 - 2\\,\\Phi(\\tfrac{-\\mu}{\\sigma}) \\right)"^^ . a ; rdfs:label "left beta parameter of Beta-binomial-1" ; "left beta parameter of Beta-binomial-1"^^ ; "BetaBinomial1.alpha"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "left beta parameter"^^ ; "alpha"^^ . a ; rdfs:label "vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-1" ; "vector of positive inverse scales of Multivariate-Gaussian-Process-Distribution-1"^^ ; "MultivariateGaussianProcess1.inverseScale"^^ ; "w"^^ ; "w \\in R^K, K \\in N"^^ ; "vector of positive inverse scales"^^ ; "inverseScale"^^ . a ; rdfs:label "mode of Gumbel 1" ; "mode of Gumbel 1"^^ ; "\\mu"^^ . a ; rdfs:label "PDF of Standard Cauchy 1" ; "PDF of Standard Cauchy 1"^^ ; "\\frac1\\pi \\frac1{1+x^2}"^^ ; "1/pi/(1+x^2)"^^ . a ; rdfs:label "lognormal"^^ , "Galton"^^ , "Log-Normal 7" ; "Log-Normal 7"^^ ; "LogNormal7"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "shape mode of Triangular-1" ; "shape (mode) of Triangular-1"^^ ; "Triangular1.shape"^^ ; "c"^^ ; "c \\in R"^^ ; "shape (mode)"^^ ; "shape"^^ . a ; rdfs:label "Power 1" ; "Power 1"^^ ; "Power1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "PDF of Logistic 2" ; "PDF of Logistic 2"^^ ; "\\frac{\\tau e^{-\\tau(x-\\mu)}} {\\left(1+e^{-\\tau(x-\\mu)}\\right)^2}"^^ ; "(tau * exp(-tau*(x-mu))) / (1+exp(-tau*(x-mu)))^2"^^ . a ; rdfs:label "CDF of Uniform 1" ; "CDF of Uniform 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}"""^^ ; "(x-a)/(b-a)"^^ . a ; rdfs:label "CDF of Negative Binomial 1" ; "CDF of Negative Binomial 1"^^ ; "1 - I_{1-p}(k+1,r)"^^ ; "1 - Rbeta(1-p, k+1, r, lower = T)"^^ . a ; rdfs:label "Standard Logistic 1" ; "Standard Logistic 1"^^ ; "StandardLogistic1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ . a ; rdfs:label "PDF of Kumaraswamy 1" ; "PDF of Kumaraswamy 1"^^ ; "a b x^{a-1} (1-x^a)^{b-1}"^^ ; "a*b*x^(a-1) * (1-x^a)^(b-1)"^^ . a ; rdfs:label "CDF of Log-Normal 6" ; "CDF of Log-Normal 6"^^ ; "\\frac12 + \\frac12\\,\\text{erf}\\Big[\\frac{\\log x-\\log m}{\\sqrt{2}\\log(\\sigma_g)}\\Big]"^^ ; "1/2 + 1/2 * erf( (log(x)-log(m)) / (sqrt(2)*log(sigma_g)) )"^^ . a ; rdfs:label "relationship between Negative Binomial 5 and Negative Binomial 4 whereby r = \\\\alpha, p = 1 / 1 + \\\\beta" ; ; ; "r = \\alpha, p = 1 / (1 + \\beta)"^^ ; "NegativeBinomial5(\\alpha, \\beta) \\rightarrow NegativeBinomial4(r,p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Student-s-t-distribution-2" ; "mean of Student-s-t-distribution-2"^^ ; "StudentT2.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "CDF of Weibull 1" ; "CDF of Weibull 1"^^ ; "1- \\exp(-(x/\\lambda)^k)"^^ ; "1- exp(-(x/lambda)^k)"^^ . a ; rdfs:label "Beta 1" ; "Beta 1"^^ ; "Beta1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,1)"^^ ; , . a ; rdfs:label "mode of Multivariate Normal 2" ; "mode of Multivariate Normal 2"^^ ; "\\mu"^^ . a ; rdfs:label "Breit-Wigner"^^ , "Standard Cauchy 1" , "Lorentz"^^ ; "Standard Cauchy 1"^^ ; "StandardCauchy1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in R"^^ . a ; rdfs:label "PDF of Hyperbolic secant 1" ; "PDF of Hyperbolic secant 1"^^ ; "\\text{sech}\\Big(\\pi x\\Big)"^^ ; "sech(pi*x)"^^ . a ; rdfs:label "CDF of Half-normal 1" ; "CDF of Half-normal 1"^^ ; "\\text{erf}(\\theta x / \\sqrt{\\pi})"^^ ; "erf(theta * x / sqrt(pi))"^^ . a ; rdfs:label "PDF of Logit Normal 1" ; "PDF of Logit Normal 1"^^ ; "\\frac{1}{\\sigma \\sqrt{2 \\pi}}\\, e^{-\\frac{(\\operatorname{logit}(x) - \\mu)^2}{2\\sigma^2}}\\frac{1}{x (1-x)}"^^ ; "1/(sigma * sqrt(2*pi)) * exp(-(logit(x) - mu)^2)/(2*sigma^2)) * 1 / (x*(1-x))"^^ . a ; rdfs:label "variance of Triangular 1" ; "variance of Triangular 1"^^ ; "(a^2 + b^2 + c^2 - ab - ac - bc)/18"^^ . a ; rdfs:label "PMF of YuleSimon1" ; "PMF of YuleSimon1"^^ ; "\\rho B(k, \\rho+1)"^^ ; "rho*beta(k,rho+1)"^^ . a ; rdfs:label "scale of Weibull-1" ; "scale of Weibull-1"^^ ; "Weibull1.scale"^^ ; "\\lambda"^^ ; "\\lambda \\in (0, +\\infty)"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "mode of Multivariate Normal 1" ; "mode of Multivariate Normal 1"^^ ; "\\mu"^^ . a ; rdfs:label "CDF of Folded Normal 1" ; "CDF of Folded Normal 1"^^ ; "\\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]"^^ ; "1/2 * (erf((x-mu)/(sigma*sqrt(2))) + erf((x+mu)/(sigma*sqrt(2))))"^^ . a owl:DatatypeProperty ; rdfs:label "has code name" . a ; rdfs:label "standard deviation of Truncated-Normal-1" ; "standard deviation of Truncated-Normal-1"^^ ; "TruncatedNormal1.stdev"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "standard deviation"^^ ; "stdev"^^ . a ; rdfs:label "PDF of Amoroso 1" ; "PDF of Amoroso 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)"^^ . a ; rdfs:label "PDF of Beta 1" ; "PDF of Beta 1"^^ ; "\\frac{x^{\\alpha-1}(1-x)^{\\beta-1}} {B(\\alpha,\\beta)} "^^ ; "(x^(alpha-1)*(1-x)^(beta-1))/beta(alpha,beta)"^^ . a ; rdfs:label "Johnson SU 1" ; "Johnson SU 1"^^ ; "JohnsonSU1"^^ ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , , , . a ; rdfs:label "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"^^ ; "Normal1(\\mu,\\sigma) \\rightarrow StandardNormal1(0,1)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalStandardnormalT.pdf}"""^^ . a ; rdfs:label "CDF of Generalized Gamma 2" ; "CDF of Generalized Gamma 2"^^ ; "\\frac{\\gamma(c,(\\frac{x-a}{b})^k)}{\\Gamma(c)}"^^ ; "Igamma(c, ((x-a)/b)^k, lower=T) / gamma(c)"^^ . a ; rdfs:label "HF of Power 1" ; "HF of Power 1"^^ ; "\\frac{\\beta x^{\\beta-1} \\alpha^{-\\beta}}{\\alpha^\\beta - x^\\beta}"^^ ; "(beta * x^(beta-1) * alpha^(-beta)) / (alpha^beta - x^beta)"^^ . a ; rdfs:label "relationship between Amoroso 1 and Frechet 1 whereby \\\\text{Frechet1}\\\\alpha,\\\\sigma \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a=0, \\\\theta=\\\\sigma, \\\\alpha=1, \\\\beta rightarrow -\\\\beta" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a owl:Class ; rdfs:label "real number" ; rdfs:subClassOf . a ; rdfs:label "CDF of Logistic 2" ; "CDF of Logistic 2"^^ ; "\\frac{1}{1+e^{-\\tau(x-\\mu)}}"^^ ; "1/(1+exp(-tau*(x-mu)))"^^ . a ; rdfs:label "shape parameter of Kumaraswamy-1" ; "shape parameter of Kumaraswamy-1"^^ ; "Kumaraswamy1.shape1"^^ ; "a"^^ ; "a > 0"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "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)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "lower limit of Triangular-1" ; "lower limit of Triangular-1"^^ ; "Triangular1.lowerLimit"^^ ; "a"^^ ; "a \\in R"^^ ; "lower limit"^^ ; "lowerLimit"^^ . a ; rdfs:label "CDF of Logit Normal 1" ; "CDF of Logit Normal 1"^^ ; "\\frac12\\Big[1 + \\operatorname{erf}\\Big( \\frac{\\operatorname{logit}(x)-\\mu}{\\sqrt{2\\sigma^2}}\\Big)\\Big]"^^ ; "1/2*(1 + erf((logit(x)-mu)/sqrt(2*sigma^2))))"^^ . a ; rdfs:label "YuleSimon1" ; "YuleSimon1"^^ ; "YuleSimon1"^^ ; ; ; ; ; ; "k"^^ ; "k \\in \\{1,2,\\dots\\}"^^ ; . a ; rdfs:label "Hyperbolic secant 1" ; "Hyperbolic secant 1"^^ ; "HyperbolicSecant1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ . a ; rdfs:label "PDF of Folded Normal 1" ; "PDF of Folded Normal 1"^^ ; "\\frac{1}{\\sigma\\sqrt{2\\pi}} \\,e^{ -\\frac{(x-\\mu)^2}{2\\sigma^2} } + \\frac{1}{\\sigma\\sqrt{2\\pi}} \\,e^{ -\\frac{(x+\\mu)^2}{2\\sigma^2} }"^^ ; "1/(sigma*sqrt(2*pi))*exp(-(x-mu)^2/(2*sigma^2)) + 1/(sigma*sqrt(2*pi))*exp(-(x+mu)^2/(2*sigma^2))"^^ . a owl:DatatypeProperty ; rdfs:label "has name" . a ; rdfs:label "index parameter of Inverse-Binomial-1" ; "index parameter of Inverse-Binomial-1"^^ ; "InverseBinomial1.index"^^ ; "k"^^ ; "k \\in \\left\\{0,1,2,\\dots\\right\\}"^^ ; "index parameter"^^ ; "index"^^ . a ; rdfs:label "relationship between Gamma 1 and Chi-squared 1 whereby k_{ChiSquared1}=2k, \\\\theta=2" ; ; ; "k_{ChiSquared1}=2k, \\theta=2"^^ ; "Gamma1(k,\\theta) \\rightarrow ChiSquared1(n)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GammaChisquareT.pdf}"^^ . a ; rdfs:label "variance of Multivariate Normal 1" ; "variance of Multivariate Normal 1"^^ ; "\\Sigma"^^ . a ; rdfs:label "variance of Weibull 1" ; "variance of Weibull 1"^^ ; "\\lambda^2\\left[\\Gamma\\left(1+\\frac{2}{k}\\right) - \\left(\\Gamma\\left(1+\\frac{1}{k}\\right)\\right)^2\\right]"^^ . a ; rdfs:label "location parameter of Amoroso-1" ; "location parameter of Amoroso-1"^^ ; "Amoroso1.location"^^ ; "a"^^ ; "a \\in R"^^ ; "location parameter"^^ ; "location"^^ . a ; rdfs:label "logit of probability of success of Bernoulli-2" ; "logit of probability of success of Bernoulli-2"^^ ; "Bernoulli2.logitProbability"^^ ; "\\alpha"^^ ; "\\alpha \\in R"^^ ; "logit of probability of success"^^ ; "logitProbability"^^ . a ; rdfs:label "mean of Truncated-Normal-1" ; "mean of Truncated-Normal-1"^^ ; "TruncatedNormal1.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "relationship between GeneralizedPoisson2 and Generalized Poisson 1 whereby \\\\theta = \\\\mu 1-\\\\delta" ; ; ; "\\theta = \\mu (1-\\delta)"^^ ; "GeneralizedPoisson2(\\mu,\\delta) \\rightarrow GeneralizedPoisson1(\\theta,\\delta)"^^ ; """\\cite{Yang:2007vn} \\\\ ProbOnto spec"""^^ . a ; rdfs:label "mode of Kumaraswamy 1" ; "mode of Kumaraswamy 1"^^ ; "\\left(\\frac{a-1}{ab-1}\\right)^{1/a} \\text{ for } a\\geq 1, b\\geq 1, (a,b)\\neq (1,1)"^^ . a ; rdfs:label "scale parameter of Amoroso-1" ; "scale parameter of Amoroso-1"^^ ; "Amoroso1.scale"^^ ; "\\theta"^^ ; "\\theta \\in R"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "mean of Logistic 2" ; "mean of Logistic 2"^^ ; "\\mu"^^ . a ; rdfs:label "relationship between Geometric 1 and Negative Binomial 1 whereby \\\\Sigma X iid" ; ; ; "\\Sigma X (iid)"^^ ; "Geometric1(p) \\rightarrow NegativeBinomial1(r,p)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/GeometricPascal.pdf}"""^^ . a ; rdfs:label "variance of Standard Logistic 1" ; "variance of Standard Logistic 1"^^ ; "\\frac{\\pi^2}{3}"^^ . a ; rdfs:label "mean of Generalized Gamma 2" ; "mean of Generalized Gamma 2"^^ ; "a + b\\Gamma(c+1/k)/\\Gamma(c)"^^ . a ; rdfs:label "CDF of Beta 1" ; "CDF of Beta 1"^^ ; "I_x(\\alpha,\\beta)"^^ ; "Rbeta(x, alpha, beta)"^^ . a owl:Class ; rdfs:label "rational number" ; rdfs:subClassOf . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Kumaraswamy_distribution}"^^ . a ; rdfs:label "CDF of Power 1" ; "CDF of Power 1"^^ ; "\\frac{x^\\beta}{\\alpha^\\beta}"^^ ; "x^beta / alpha^beta"^^ . a ; rdfs:label "mean of Triangular 1" ; "mean of Triangular 1"^^ ; "(a+b+c)/3"^^ . a ; rdfs:label "mean of YuleSimon1" ; "mean of YuleSimon1"^^ ; "\\frac{\\rho}{\\rho-1} \\text{ for } \\rho>1"^^ . a ; rdfs:label "mean of Logit Normal 1" ; "mean of Logit Normal 1"^^ ; "\\text{ no analytical solution }"^^ . a ; rdfs:label "variance of Half-normal 1" ; "variance of Half-normal 1"^^ ; "\\frac{\\pi - 2}{2 \\theta^2}"^^ . a ; rdfs:label "variance of Truncated Normal 1" ; "variance of Truncated Normal 1"^^ ; """\\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]"""^^ . a ; rdfs:label "relationship between Truncated Normal 1 and Normal 1 whereby a=-\\\\infty, b=\\\\infty" ; ; ; "a=-\\infty, b=\\infty"^^ ; "TruncatedNormal1(\\mu,\\sigma,a,b) \\rightarrow Normal1(\\mu,\\sigma)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "CDF of Trapezoidal 1" ; "CDF of Trapezoidal 1"^^ ; """\\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 \\\\"""^^ ; """CDF1 = 1/(d+c-a-b)*1/(b-a)*(x-a)^2 CDF2 = 1/(d+c-a-b)*(2*x-a-b) CDF3 = 1-1/(d+c-a-b)*1/(d-c)*(d-x)^2"""^^ . a ; rdfs:label "precision matrix of Multivariate-Normal-2" ; "precision matrix of Multivariate-Normal-2"^^ ; "MultivariateNormal2.precisionMatrix"^^ ; "T"^^ ; "\\text{inverse of the covariance matrix}"^^ ; "precision matrix"^^ ; "precisionMatrix"^^ . a ; rdfs:label "Gumbel 1" ; "Gumbel 1"^^ ; "Gumbel1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "CDF of Multivariate Normal 1" ; "CDF of Multivariate Normal 1"^^ ; "\\text{no analytic expression}"^^ . a ; rdfs:label "mode of Weibull 1" ; "mode of Weibull 1"^^ ; """\\begin{cases} \\lambda \\left(\\frac{k-1}{k} \\right)^{\\frac{1}{k}}\\, &k>1\\\\ 0 &k=1\\end{cases}"""^^ . a ; rdfs:label "mode of Hypergeometric 1" ; "mode of Hypergeometric 1"^^ ; "\\left \\lfloor \\frac{(n+1)(K+1)}{N+2} \\right \\rfloor"^^ . a owl:Class ; rdfs:label "hazard function" ; rdfs:subClassOf . a ; rdfs:label "relationship between Negative Binomial 1 and Negative Binomial 2 whereby \\\\tau = 1/r, \\\\lambda = r1-p/p" ; ; ; "\\tau = 1/r, \\lambda = r(1-p)/p"^^ ; "NegativeBinomial1(r, p) \\rightarrow NegativeBinomial2(\\lambda, \\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Folded Normal 1" ; "Folded Normal 1"^^ ; "FoldedNormal1"^^ ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a owl:DatatypeProperty ; rdfs:label "has alternate name" . a ; rdfs:label "mode of Standard Logistic 1" ; "mode of Standard Logistic 1"^^ ; "0"^^ . a ; rdfs:label "Poisson intensity of Negative-Binomial-2" ; "Poisson intensity of Negative-Binomial-2"^^ ; "NegativeBinomial2.rate"^^ ; "\\lambda"^^ ; "\\lambda \\in R, \\lambda > 0"^^ ; "Poisson intensity"^^ ; "rate"^^ . a ; rdfs:label "median of Logistic 2" ; "median of Logistic 2"^^ ; "\\mu"^^ . a ; rdfs:label "median of Kumaraswamy 1" ; "median of Kumaraswamy 1"^^ ; "\\left(1-2^{-1/b}\\right)^{1/a}"^^ . a ; rdfs:label "relationship between Gamma 2 and Gamma 1 whereby k=r, \\\\theta = 1 / \\\\mu" ; ; ; "k=r, \\theta = 1 / \\mu"^^ ; "Gamma2(r,mu) \\rightarrow Gamma1(k,\\theta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "relationship between Student's t-distribution 3 and Student's t-distribution 2 whereby \\\\tau=1/\\\\sigma^2" ; ; ; "\\tau=1/\\sigma^2"^^ ; "StudentT3(\\nu,\\mu,\\sigma) \\rightarrow StudentT2(\\mu,\\tau,k)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "relationship between Amoroso 1 and Weibull 1 whereby \\\\text{Weibull1}\\\\lambda,k \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a = 0, \\\\theta=\\\\lambda, \\\\alpha=1" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution}"^^ . a ; rdfs:label "Generalized Gamma 2" ; "Generalized Gamma 2"^^ ; "GeneralizedGamma2"^^ ; ; ; ; ; ; "x"^^ ; "0 < a < x"^^ ; , , , . a ; rdfs:label "inverse scale of Half-normal-1" ; "inverse scale of Half-normal-1"^^ ; "HalfNormal1.inverseScale"^^ ; "\\theta"^^ ; "\\theta > 0"^^ ; "inverse scale"^^ ; "inverseScale"^^ . a ; rdfs:label "mean of Power 1" ; "mean of Power 1"^^ ; "\\frac{\\alpha \\beta}{1+\\beta}"^^ . a ; rdfs:label "NA of Makeham-1" ; "NA of Makeham-1"^^ ; "Makeham1.delta"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; "delta"^^ . a ; rdfs:label "CDF of YuleSimon1" ; "CDF of YuleSimon1"^^ ; "1 - kB(k, \\rho+1)"^^ ; "1 - k*beta(k,rho+1)"^^ . a ; rdfs:label "mode of Triangular 1" ; "mode of Triangular 1"^^ ; "c"^^ . a ; rdfs:label "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)"^^ ; """ProbOnto spec \\\\ \\cite{cameron2013regression}"""^^ . a ; rdfs:label "mean of Truncated Normal 1" ; "mean of Truncated Normal 1"^^ ; "\\mu + \\frac{ \\phi(\\frac{a-\\mu}{\\sigma}) - \\phi(\\frac{b-\\mu}{\\sigma}) }{ \\Phi(\\frac{b-\\mu}{\\sigma}) - \\Phi(\\frac{a-\\mu}{\\sigma}) } \\sigma"^^ . a ; rdfs:label "mean of Half-normal 1" ; "mean of Half-normal 1"^^ ; "1/\\theta"^^ . a ; rdfs:label "median of Weibull 1" ; "median of Weibull 1"^^ ; "\\lambda(\\log(2))^{1/k}"^^ . a ; rdfs:label "mean of Multivariate Normal 1" ; "mean of Multivariate Normal 1"^^ ; "\\mu"^^ . a ; rdfs:label "shape of Standard-Two-Sided-Power-1" ; "shape of Standard-Two-Sided-Power-1"^^ ; "StandardTwoSidedPower1.shape"^^ ; "n"^^ ; "n > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label """relationship between Log-Normal 6 and Log-Normal 2 whereby \\\\mu=\\\\logm, v=\\\\log\\\\sigma_g^2""" ; ; ; """\\mu=\\log(m), v=\\log(\\sigma_g^2)"""^^ ; "LogNormal6(m,\\sigma_g) \\rightarrow LogNormal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "scale of Gamma-1" ; "scale of Gamma-1"^^ ; "Gamma1.scale"^^ ; "\\theta"^^ ; "\\theta > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "Amoroso 1" ; "Amoroso 1"^^ ; "Amoroso1"^^ ; ; "x"^^ ; "x \\geq a \\text{ if } \\theta > 0, x \\leq a \\text{ if } \\theta < 0"^^ ; , , , . a ; rdfs:label "PDF of Gumbel 1" ; "PDF of Gumbel 1"^^ ; "\\frac{e^{-e^{-\\frac{x-\\mu}{\\beta}}} e^{-\\frac{x-\\mu}{\\beta}}}{\\beta}"^^ ; "(exp(-exp(-(x-mu)/beta)) * exp(-(x-mu)/beta))/beta"^^ . a owl:Class ; rdfs:label "survival function" ; rdfs:subClassOf . a ; rdfs:label "median of Logit Normal 1" ; "median of Logit Normal 1"^^ ; "\\frac{\\exp(\\mu)}{1+\\exp(\\mu)}"^^ . a ; rdfs:label "median of Standard Logistic 1" ; "median of Standard Logistic 1"^^ ; "0"^^ . a owl:DatatypeProperty ; rdfs:label "has abbreviation" . a ; rdfs:label "variance of Log-Normal 6" ; "variance of Log-Normal 6"^^ ; "m^2 e^{\\log^2(\\sigma_g)} [e^{\\log^2(\\sigma_g)}-1]"^^ . a ; rdfs:label "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"^^ ; "\\url{https://en.wikipedia.org/wiki/Kumaraswamy_distribution}"^^ . a ; rdfs:label "mean of Kumaraswamy 1" ; "mean of Kumaraswamy 1"^^ ; "\\frac{b\\Gamma(1+\\tfrac{1}{a})\\Gamma(b)}{\\Gamma(1+\\tfrac{1}{a}+b)}"^^ . a ; rdfs:label "mode of Logistic 2" ; "mode of Logistic 2"^^ ; "\\mu"^^ . a ; rdfs:label "relationship between Beta Negative Binomial1 and Negative Binomial 1 whereby \\\\text{If } X \\\\sim \\\\text{NegativeBinomial1n,p and } p \\\\sim \\\\text{Beta1}\\\\alpha,\\\\beta \\\\text{ then } X \\\\sim \\\\text{NegativeBinomial1}\\\\alpha,\\\\beta,n" ; ; ; "\\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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/PascalBetapascal.pdf}"^^ . a ; rdfs:label "relationship between Beta-binomial 1 and Uniform Discrete 2 whereby \\\\alpha=1, \\\\beta=1" ; ; ; "\\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}"""^^ . a ; rdfs:label """relationship between Conway-Maxwell-Poisson 1 and Negative Binomial 1 whereby \\\\text{ For } \\\\nu=0 \\\\text{ and } \\\\lambda<1 \\\\text{ the sum of Conway-Maxwell-Poisson distributed variables}\\\\\\\\ \\\\text{reduces to the sum of geometric variables, which follows a Negative Binomial distribution with parameters}\\\\\\\\ n \\\\text{ and } 1-\\\\lambda""" ; ; ; """\\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)"^^ ; "\\cite{shmueli2005useful}"^^ . a ; rdfs:label "PDF of Generalized Gamma 2" ; "PDF of Generalized Gamma 2"^^ ; "\\frac{k (x-a)^{kc-1}}{b^{kc}\\Gamma(c)}\\exp\\Big[-\\Big(\\frac{x-a}{b}\\Big)^k\\Big]"^^ ; "(k*(x-a)^(k*c-1)) / (b^(k*c)*gamma(c)) * exp( -((x-a)/b)^k )"^^ . a ; rdfs:label "variance of Makeham 1" ; "variance of Makeham 1"^^ ; "\\text{mathematically intractable}"^^ . a ; rdfs:label "SF of Power 1" ; "SF of Power 1"^^ ; "1- \\frac{x^\\beta}{\\alpha^\\beta}"^^ ; "1- x^beta / alpha^beta"^^ . a ; rdfs:label "variance of Uniform 1" ; "variance of Uniform 1"^^ ; "\\tfrac{1}{12} (b-a)^2"^^ . a ; rdfs:label "PDF of Muth 1" ; "PDF of Muth 1"^^ ; "(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 )"^^ . a ; rdfs:label "HF of Arcsine 1" ; "HF of Arcsine 1"^^ ; "\\frac{2}{\\sqrt{x(1-x)} (\\pi - 2 \\arcsin\\left(2x -1\\right))}"^^ ; "2/(sqrt(x*(1-x))*(pi - 2 * asin(2*x -1)))"^^ . a ; rdfs:label "Power-Normal 1" ; "Power-Normal 1"^^ ; "PowerNormal1"^^ ; ; ; "x"^^ ; "x \\in R"^^ ; , , . a ; rdfs:label "PDF of Wishart 2" ; "PDF of Wishart 2"^^ ; "|R|^{k/2}|x|^{(k-p-1)/2}e^{-\\frac{1}2\\,tr(Rx)}"^^ . a ; rdfs:label "PMF of Zero-inflated Poisson 1" ; "PMF of Zero-inflated Poisson 1"^^ ; """\\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"""^^ . a ; rdfs:label "PDF of Dagum 1" ; "PDF of Dagum 1"^^ ; "\\frac{a p}{x} \\left( \\frac{(\\tfrac{x}{b})^{a p}}{\\left((\\tfrac{x}{b})^a + 1 \\right)^{p+1}} \\right)"^^ ; "a*p/x * (x/b)^(a*p) / ((x/b)^a + 1)^(p+1)"^^ . a ; rdfs:label "scale of Cauchy-1" ; "scale of Cauchy-1"^^ ; "Cauchy1.scale"^^ ; "\\gamma"^^ ; "\\gamma \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "relationship between Log-Normal 4 and Log-Normal 2 whereby \\\\mu = \\\\logm, v = \\\\logcv^2+1" ; ; ; "\\mu = \\log(m), v = \\log(cv^2+1)"^^ ; "LogNormal4(m,cv) \\rightarrow LogNormal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Uniform Discrete 2" , "Rectangular Discrete 2"^^ ; "Uniform Discrete 2"^^ ; "UniformDiscrete2"^^ ; ; ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,\\dots,n\\}"^^ ; . a ; rdfs:label "scale parameter of Johnson-SB-1" ; "scale parameter of Johnson-SB-1"^^ ; "JohnsonSB1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "mean of Noncentral F 1" ; "mean of Noncentral F 1"^^ ; "\\frac{n}{m} \\frac{m+lambda}{n-2}"^^ . a ; rdfs:label "PMF of Poisson 1" ; "PMF of Poisson 1"^^ ; "\\frac{\\lambda^k}{k!}e^{-\\lambda}"^^ ; "lambda^k/factorial(k) * exp(-lambda)"^^ . a ; rdfs:label "mean of Von Mises 1" ; "mean of Von Mises 1"^^ ; "\\mu"^^ . a ; rdfs:label "Birnbaum-Saunders 1" ; "Birnbaum-Saunders 1"^^ ; "BirnbaumSaunders1"^^ ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "median of Gompertz 1" ; "median of Gompertz 1"^^ ; "\\left(1/b\\right)\\log\\left[\\left(-1/\\eta\\right) \\log\\left(1/2\\right)+1\\right]"^^ . a ; rdfs:label "relationship between Log-Normal 5 and Log-Normal 1 whereby \\\\mu = \\\\mu, \\\\sigma = 1 / \\\\sqrt{\\\\tau}" ; ; ; "\\mu = \\mu, \\sigma = 1 / \\sqrt{\\tau}"^^ ; "LogNormal5(\\mu,\\tau) \\rightarrow LogNormal1(\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Poisson 2" ; "CDF of Poisson 2"^^ ; "\\frac{\\gamma(\\lfloor k+1 \\rfloor,\\exp(\\alpha))}{\\lfloor k \\rfloor!}"^^ ; "Igamma(floor(k+1), exp(alpha), lower=F) / factorial(floor(k))"^^ . a ; rdfs:label "mean of Inverse Binomial 1" ; "mean of Inverse Binomial 1"^^ ; "k (1-p) / (2p-1)"^^ . a ; rdfs:label "Logit Normal 1" ; "Logit Normal 1"^^ ; "LogitNormal1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,1)"^^ ; , . a ; rdfs:label "location of Cauchy-1" ; "location of Cauchy-1"^^ ; "Cauchy1.location"^^ ; "x_0"^^ ; "x_0 \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "PDF of Sinh-Arcsinh 1" ; "PDF of Sinh-Arcsinh 1"^^ ; """\\frac{c}{\\sqrt{2\\pi} \\sigma (1+z^2)^{1/2}} e^{-r^2/2}\\\\ r = \\frac12 \\left(\\exp[\\tau \\sinh^{-1}(z)] - \\exp[-\\nu \\sinh^{-1}(z)]\\right)\\\\ c = \\frac12 \\left( \\tau \\exp[\\tau \\sinh^{-1}(z)] + \\nu \\exp[-\\nu \\sinh^{-1}(z)] \\right)\\\\ z = (x-\\mu)/\\sigma"""^^ ; """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 }"""^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Categorical Ordered 1" ; "mean of Categorical Ordered 1"^^ ; "E([x=i]) = p_i, \\text{this is the mean of the Iverson bracket } [x=i] \\text{ and not the mean of } x"^^ . a ; rdfs:label "mean of Uniform Discrete 1" ; "mean of Uniform Discrete 1"^^ ; "(a+b)/2"^^ . a ; rdfs:label "scale of Pareto-Type-II" ; "scale of Pareto-Type-II"^^ ; "ParetoTypeII1.scale"^^ ; "\\lambda"^^ ; "\\lambda \\in R^+"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "CDF of Arcsine 1" ; "CDF of Arcsine 1"^^ ; "\\frac{2}{\\pi}\\arcsin\\left(\\sqrt x \\right)"^^ ; "2/pi*arcsin(sqrt(x))"^^ . a ; rdfs:label "Erlang 1" ; "Erlang 1"^^ ; "Erlang1"^^ ; ; ; ; ; ; ; ; "x"^^ ; """x \\in [0,+\\infty) \\text{ (Forbes)}, x \\in (0,+\\infty) \\text{ (Leemis)}"""^^ ; , . a ; rdfs:label "Zero-inflated Poisson 1" ; "Zero-inflated Poisson 1"^^ ; "ZeroInflatedPoisson1"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "Poisson 1" ; "Poisson 1"^^ ; "Poisson1"^^ ; ; ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; . a ; rdfs:label "PDF of Power-Normal 1" ; "PDF of Power-Normal 1"^^ ; """\\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)) } }"""^^ . a ; rdfs:label "location parameter of Johnson-SB-1" ; "location parameter of Johnson-SB-1"^^ ; "JohnsonSB1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location parameter"^^ ; "location"^^ . a ; rdfs:label "relationship between Generalized Poisson 1 and GeneralizedPoisson2 whereby \\\\mu = \\\\theta / 1-\\\\delta" ; ; ; "\\mu = \\theta / (1-\\delta)"^^ ; "GeneralizedPoisson1(\\theta,\\delta) \\rightarrow GeneralizedPoisson2(\\mu,\\delta)"^^ ; """\\cite{Yang:2007vn} \\\\ ProbOnto spec"""^^ . a ; rdfs:label "CDF of Noncentral F 1" ; "CDF of Noncentral F 1"^^ ; """\\int_0^x f(x), \\text { with f the PDF}"""^^ ; "cumsum(PDF*rep(stepSize,length(PDF)))"^^ . a ; rdfs:label "PDF of Student's t-distribution 2" ; "PDF of Student's t-distribution 2"^^ ; "\\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}}"^^ ; "gamma((k+1)/2)/gamma(k/2)*sqrt(tau/(k*pi))*(1+tau/k*(x-mu)^2)^(-(k+1)/2)"^^ . a ; rdfs:label "Double Exponential 2"^^ , "Laplace 2" ; "Laplace 2"^^ ; "Laplace2"^^ ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "CDF of Muth 1" ; "CDF of Muth 1"^^ ; "1-e^{\\left[-\\frac{e^{\\kappa x}}{\\kappa} + \\kappa x + \\frac1{\\kappa} \\right]}"^^ ; "1-exp( -exp(kappa*x)/x + kappa*x + 1/kappa )"^^ . a ; rdfs:label "PDF of Birnbaum-Saunders 1" ; "PDF of Birnbaum-Saunders 1"^^ ; "\\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) """^^ . a ; rdfs:label "CDF of Von Mises 1" ; "CDF of Von Mises 1"^^ ; "\\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) }"""^^ . a ; rdfs:label "concentration of Dirichlet-1" ; "concentration of Dirichlet-1"^^ ; "Dirichlet1.concentration"^^ ; "\\alpha_1, \\cdots, \\alpha_K"^^ ; "\\alpha_1, \\cdots, \\alpha_K, \\alpha_i > 0"^^ ; "concentration"^^ ; "concentration"^^ . a ; rdfs:label "SF of Gompertz 1" ; "SF of Gompertz 1"^^ ; "\\exp\\left(-\\eta\\left(e^{bx}-1 \\right)\\right)"^^ ; "exp(-eta*(exp(b*x)-1))"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "location of Sinh-Arcsinh-1" ; "location of Sinh-Arcsinh-1"^^ ; "SinhArcsinh1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "relationship between Trapezoidal 1 and Triangular 1 whereby b=c" ; ; ; "b=c"^^ ; "Trapezoidal1(a,b,c,d) \\rightarrow Triangular1(a,b,c)"^^ ; "\\cite{Garvey:2000yq}"^^ . a ; rdfs:label "PMF of Poisson 2" ; "PMF of Poisson 2"^^ ; "\\frac1{k!}\\exp\\big(k \\alpha - \\exp(\\alpha)\\big)"^^ ; "1/factorial(k) * exp(k*alpha - exp(alpha))"^^ . a ; rdfs:label "variance of Inverse Binomial 1" ; "variance of Inverse Binomial 1"^^ ; "k p (1-p) / (2p - 1)^3"^^ . a ; rdfs:label "CDF of Categorical Ordered 1" ; "CDF of Categorical Ordered 1"^^ ; """\\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}"""^^ . a ; rdfs:label "tail index of Pareto-Type-II" ; "tail index of Pareto-Type-II"^^ ; "ParetoTypeII1.tailIndex"^^ ; "\\alpha"^^ ; "\\alpha > 0, \\alpha \\in R"^^ ; "tail index"^^ ; "tailIndex"^^ . a ; rdfs:label "CDF of Uniform Discrete 2" ; "CDF of Uniform Discrete 2"^^ ; "\\frac{k+1}{n+1}"^^ ; "(k+1)/(n+1)"^^ . a ; rdfs:label "relationship between Zero-Inflated Negative Binomial 1 and Zero-inflated Poisson 1 whereby \\\\tau \\\\rightarrow 0" ; ; ; "\\tau \\rightarrow 0"^^ ; "ZeroInflatedNegativeBinomial1(\\lambda,\\tau,p0) \\rightarrow ZeroInflatedPoisson1(\\lambda,\\pi)"^^ ; "\\cite{Troconiz:2009fv}"^^ . a ; rdfs:label "PDF of Chi-squared 1" ; "PDF of Chi-squared 1"^^ ; "\\frac{1}{2^{\\frac{k}{2}}\\Gamma\\left(\\frac{k}{2}\\right)}\\; x^{\\frac{k}{2}-1} e^{-\\frac{x}{2}}"^^ ; "1/( 2^k/2 * gamma(k/2) ) * x^(k/2-1) * exp(-x/2)"^^ . a ; rdfs:label "overdispersion of Negative-Binomial-2" ; "overdispersion of Negative-Binomial-2"^^ ; "NegativeBinomial2.overdispersion"^^ ; "\\tau"^^ ; "\\tau \\in R"^^ ; "overdispersion"^^ ; "overdispersion"^^ . a ; rdfs:label "inverse scale of Negative-Binomial-5" ; "inverse scale of Negative-Binomial-5"^^ ; "NegativeBinomial5.inverseScale"^^ ; "\\beta"^^ ; "\\beta \\in R^+"^^ ; "inverse scale"^^ ; "inverseScale"^^ . a ; rdfs:label "relationship between Generalized Negative Binomial 1 and Negative Binomial 4 whereby \\\\beta=1 \\\\text{ and set } m=r, \\\\theta=p" ; ; ; "\\beta=1 \\text{ and set } m=r, \\theta=p"^^ ; "GeneralizedNegativeBinomial1(\\theta,\\beta,m) \\rightarrow NegativeBinomial4(r,p)"^^ ; "\\cite{Consul:2006qf}"^^ . a ; rdfs:label "location of Gumbel-1" ; "location of Gumbel-1"^^ ; "Gumbel1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "relationship between Generalized Gamma 2 and Chi-squared 1 whereby a=0, b=2, c=k_{ChiSquare1}/2,k=1" ; ; ; "a=0, b=2, c=k_{ChiSquare1}/2,k=1"^^ ; "GeneralizedGamma2(a,b,c,k) \\rightarrow ChiSquared1(k)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "PDF of Von Mises 1" ; "PDF of Von Mises 1"^^ ; "\\frac{\\exp(\\kappa \\cos(x-\\mu))}{2\\pi I_0(\\kappa)}"^^ ; "exp(kappa * cos(x-mu)) / (2*pi * besselI(kappa, 0, expon.scaled = FALSE))"^^ . a ; rdfs:label "relationship between Log-Logistic 2 and Log-Logistic 1 whereby \\\\alpha=1/\\\\lambda, \\\\beta=\\\\kappa" ; ; ; "\\alpha=1/\\lambda, \\beta=\\kappa"^^ ; "LogLogistic2(\\lambda,\\kappa) \\rightarrow LogLogistic1(\\alpha,\\beta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape parameter of Johnson-SB-1" ; "shape parameter of Johnson-SB-1"^^ ; "JohnsonSB1.shape2"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "HF of Muth 1" ; "HF of Muth 1"^^ ; "e^{\\kappa x} - \\kappa"^^ ; "exp(kappa*x) - kappa"^^ . a ; rdfs:label "dispersion of Zero-Inflated-Generalized-Poisson-1" ; "dispersion of Zero-Inflated-Generalized-Poisson-1"^^ ; "ZeroInflatedGeneralizedPoisson1.dispersion"^^ ; "\\alpha"^^ ; "\\alpha > -1, \\alpha \\in R"^^ ; "dispersion"^^ ; "dispersion"^^ . a ; rdfs:label "mean of Poisson 1" ; "mean of Poisson 1"^^ ; "\\lambda"^^ . a ; rdfs:label "PDF of Noncentral F 1" ; "PDF of Noncentral F 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) }"""^^ . a ; rdfs:label "mode of Student's t-distribution 2" ; "mode of Student's t-distribution 2"^^ ; "\\mu \\text{ for } k > 1"^^ . a ; rdfs:label """relationship between Log-Normal 3 and Log-Normal 4 whereby m = m, cv = \\\\sqrt{\\\\exp\\\\sigma^2 - 1}""" ; ; ; """m = m, cv = \\sqrt{\\exp(\\sigma^2) - 1}"""^^ ; "LogNormal3(m,\\sigma) \\rightarrow LogNormal4(m,cv)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PMF of Inverse Binomial 1" ; "PMF of Inverse Binomial 1"^^ ; "\\frac{k \\; \\Gamma(2x + k)}{\\Gamma(x+1) \\;\\Gamma(x + k + 1)} \\; p^{k+x} (1-p)^x "^^ ; "(k * gamma(2*x+k)) / (gamma(x+1) * gamma(x+k+1)) * p^(x+k) * (1-p)^x"^^ . a ; rdfs:label "variance of Negative Binomial 3" ; "variance of Negative Binomial 3"^^ ; "\\mu + \\mu^2 / \\phi"^^ . a ; rdfs:label "HF of Gompertz 1" ; "HF of Gompertz 1"^^ ; "b\\eta\\exp(-b x)"^^ ; "b *eta * exp(-b*x)"^^ . a ; rdfs:label "variance of Dirichlet 1" ; "variance of Dirichlet 1"^^ ; "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)"^^ . a ; rdfs:label "location of Multivariate-Student-T-2" ; "location of Multivariate-Student-T-2"^^ ; "MultivariateStudentT2.mean"^^ ; "\\mu"^^ ; "\\mu = [\\mu_1, \\dots, \\mu_d]^T, \\mu_i \\in R"^^ ; "location"^^ ; "mean"^^ . a ; rdfs:label "relationship between F 1 and Chi-squared 1 whereby \\\\text{If } X \\\\sim F1n_1,n_2 \\\\text{, the limiting distribution of } n_1X \\\\text{ as } n_2 \\\\rightarrow \\\\infty \\\\text{ is} \\\\\\\\ \\\\text{ the chi-square distribution with } n_1 \\\\text{ degrees of freedom}" ; ; ; "\\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}"^^ ; "F1(n_1,n_2) \\rightarrow ChiSquared1(n)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/FChisquare.pdf}"""^^ . a ; rdfs:label "variance of Uniform Discrete 1" ; "variance of Uniform Discrete 1"^^ ; "\\frac{(b-a+1)^2 -1}{12}"^^ . a ; rdfs:label "PMF of Categorical Ordered 1" ; "PMF of Categorical Ordered 1"^^ ; "p(x=i)=p_i"^^ . a ; rdfs:label "degrees of freedom of Inverse-Chi-Square" ; "degrees of freedom of Inverse-Chi-Square"^^ ; "InverseChiSquare1.degreesOfFreedom"^^ ; "\\nu"^^ ; "\\nu \\in R^+"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "Wishart 2" ; "Wishart 2"^^ ; "Wishart2"^^ ; ; "X"^^ ; "X(p \\times p) - \\text{symmetric, positive definite matrix}"^^ ; , . a ; rdfs:label "shape of Negative-Binomial-5" ; "shape of Negative-Binomial-5"^^ ; "NegativeBinomial5.shape"^^ ; "\\alpha"^^ ; "\\alpha \\in R^+"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "Von Mises 1" ; "Von Mises 1"^^ ; "VonMises1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in \\text{any interval of length } 2\\pi"^^ ; , . a ; rdfs:label "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}"""^^ . a ; rdfs:label "Noncentral F 1" ; "Noncentral F 1"^^ ; "NoncentralF1"^^ ; ; ; ; ; "F'=\\frac{x_1/m}{x_2/n} \\text{ with } x_1 \\sim ChiSquared1(m), x_2 \\sim ChiSquared1(n)"^^ ; "F' \\in (0,+\\infty)"^^ ; , , . a ; rdfs:label "variance of Gumbel 1" ; "variance of Gumbel 1"^^ ; "\\frac{\\pi^2}{6} \\beta^2"^^ . a ; rdfs:label "relationship between Negative Binomial 3 and Negative Binomial 5 whereby \\\\alpha = \\\\phi, \\\\beta = \\\\phi / \\\\mu" ; ; ; "\\alpha = \\phi, \\beta = \\phi / \\mu"^^ ; "NegativeBinomial3(\\mu, \\phi) \\rightarrow NegativeBinomial5(\\alpha, \\beta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "probability of zero of Zero-Inflated-Generalized-Poisson-1" ; "probability of zero of Zero-Inflated-Generalized-Poisson-1"^^ ; "ZeroInflatedGeneralizedPoisson1.probabilityOfZero"^^ ; "p0"^^ ; "0 ; "probability of zero"^^ ; "probabilityOfZero"^^ . a ; rdfs:label "shape parameter of Johnson-SB-1" ; "shape parameter of Johnson-SB-1"^^ ; "JohnsonSB1.shape1"^^ ; "\\gamma"^^ ; "\\gamma \\in R"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "shape parameter of Johnson-SN-1" ; "shape parameter of Johnson-SN-1"^^ ; "JohnsonSN1.shape1"^^ ; "\\gamma"^^ ; "\\gamma \\in R"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "relationship between Poisson 1 and Poisson 2 whereby \\\\alpha = \\\\log\\\\lambda" ; ; ; "\\alpha = \\log(\\lambda)"^^ ; "Poisson1(\\lambda) \\rightarrow Poisson2(\\alpha)"^^ ; "\\cite{stan-manual:2015b}"^^ . a ; rdfs:label "SF of Muth 1" ; "SF of Muth 1"^^ ; "e^{\\left[-\\frac{e^{\\kappa x}}{\\kappa} + \\kappa x + \\frac1{\\kappa} \\right]}"^^ ; "exp( -exp(kappa*x)/x + kappa*x + 1/kappa )"^^ . a ; rdfs:label "mean of Student's t-distribution 2" ; "mean of Student's t-distribution 2"^^ ; "\\mu"^^ . a ; rdfs:label "CDF of Poisson 1" ; "CDF of Poisson 1"^^ ; "\\frac{\\gamma(\\lfloor k+1 \\rfloor,\\lambda)}{\\lfloor k \\rfloor!}"^^ ; "Igamma(floor(k+1), lambda, lower=F) / factorial(floor(k))"^^ . a ; rdfs:label "PMF of Uniform Discrete 2" ; "PMF of Uniform Discrete 2"^^ ; "1/(n+1)"^^ ; "1/(n+1)"^^ . a ; rdfs:label "variance of Cauchy 1" ; "variance of Cauchy 1"^^ ; "undefined"^^ . a ; rdfs:label "CDF of Gompertz 1" ; "CDF of Gompertz 1"^^ ; "1-\\exp\\left(-\\eta\\left(e^{bx}-1 \\right)\\right)"^^ ; "1-exp(-eta*(exp(b*x)-1))"^^ . a ; rdfs:label "relationship between Negative Binomial 6 and Negative Binomial 3 whereby \\\\mu = \\\\exp\\\\eta" ; ; ; "\\mu = \\exp(\\eta)"^^ ; "NegativeBinomial6(\\eta,\\phi) \\rightarrow NegativeBinomial3(\\mu,\\phi)"^^ ; "\\cite{stan-manual:2015b}"^^ . a ; rdfs:label "scale parameter of Johnson-SL-1" ; "scale parameter of Johnson-SL-1"^^ ; "JohnsonSL1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "relationship between GeneralizedPoisson2 and Poisson 1 whereby \\\\delta = 0, \\\\lambda=\\\\mu" ; ; ; "\\delta = 0, \\lambda=\\mu"^^ ; "GeneralizedPoisson2(\\mu,\\delta) \\rightarrow Poisson1(\\lambda)"^^ ; "\\cite{Plan:2014kq}"^^ . a ; rdfs:label "median of Uniform Discrete 1" ; "median of Uniform Discrete 1"^^ ; "(a+b)/2"^^ . a ; rdfs:label """relationship between Normal 2 and Normal 1 whereby \\\\mu = \\\\mu, \\\\sigma = \\\\sqrt{v}""" ; ; ; """\\mu = \\mu, \\sigma = \\sqrt{v}"""^^ ; "Normal2(\\mu,v) \\rightarrow Normal1(\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "Categorical Ordered 1" ; "Categorical Ordered 1"^^ ; "CategoricalOrdered1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in \\{1,\\dots,k\\}"^^ ; . a ; rdfs:label "Sinh-Arcsinh 1" , "SHASH"^^ ; "Sinh-Arcsinh 1"^^ ; "SinhArcsinh1"^^ ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , , , . a ; rdfs:label "Binomial 1" ; "Binomial 1"^^ ; "Binomial1"^^ ; ; ; ; ; ; ; "k"^^ ; "k \\in \\{0,\\dots,n\\}"^^ ; , . a ; rdfs:label "shape parameter of Johnson-SN-1" ; "shape parameter of Johnson-SN-1"^^ ; "JohnsonSN1.shape2"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "mean of Muth 1" ; "mean of Muth 1"^^ ; "1"^^ . a ; rdfs:label "CDF of Johnson SB 1" ; "CDF of Johnson SB 1"^^ ; """\\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)))"""^^ . a ; rdfs:label "mode of Poisson 1" ; "mode of Poisson 1"^^ ; "\\lceil\\lambda\\rceil - 1, \\lfloor\\lambda\\rfloor"^^ . a ; rdfs:label "probability of extra zeros of Zero-inflated-Poisson-1" ; "probability of extra zeros of Zero-inflated-Poisson-1"^^ ; "ZeroInflatedPoisson1.probabilityOfZero"^^ ; "\\pi"^^ ; "0<\\pi<1, \\pi \\in R"^^ ; "probability of extra zeros"^^ ; "probabilityOfZero"^^ . a ; rdfs:label "location of Power-Normal-1" ; "location of Power-Normal-1"^^ ; "PowerNormal1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "variance of Negative Binomial 5" ; "variance of Negative Binomial 5"^^ ; "\\alpha/\\beta^2 (\\beta+1)"^^ . a ; rdfs:label "SF of Uniform Discrete 2" ; "SF of Uniform Discrete 2"^^ ; "\\frac{n-k}{n+1}"^^ ; "(n-k)/(n+1)"^^ . a ; rdfs:label "mode of Inverse Chi-Square" ; "mode of Inverse Chi-Square"^^ ; "\\frac{1}{\\nu + 2} "^^ . a ; rdfs:label "relationship between Standard Normal 1 and Chi-squared 1 whereby \\\\Sigma X^2" ; ; ; "\\Sigma X^2"^^ ; "StandardNormal1(0,1) \\rightarrow ChiSquared1(k)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandardnormalChisquare.pdf}"""^^ . a ; rdfs:label "mean of Negative Binomial 5" ; "mean of Negative Binomial 5"^^ ; "\\alpha / \\beta"^^ . a ; rdfs:label "logit of success probability in each trial of Binomial-2" ; "logit of success probability in each trial of Binomial-2"^^ ; "Binomial2.logitProbability"^^ ; "\\alpha"^^ ; "\\alpha \\in R"^^ ; "logit of success probability in each trial"^^ ; "logitProbability"^^ . a ; rdfs:label "mean of logx of Log-Normal-1" ; "mean of log(x) of Log-Normal-1"^^ ; "LogNormal1.meanLog"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean of log(x)"^^ ; "meanLog"^^ . a ; rdfs:label """relationship between Log-Normal 6 and Log-Normal 3 whereby m = m, \\\\sigma=\\\\log\\\\sigma_g""" ; ; ; """m = m, \\sigma=\\log(\\sigma_g)"""^^ ; "LogNormal6(m,\\sigma_g) \\rightarrow LogNormal3(m,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PDF of Multivariate Student T 2" ; "PDF of Multivariate (Student) T 2"^^ ; "\\frac{\\Gamma((k+d)/2)}{\\Gamma(k/2) k^{d/2} \\pi^{d/2}}|T|^{1/2} \\Big[ 1 + \\frac{1}{k}(x-\\mu)' T (x-\\mu) \\Big]^{-(k+d)/2}"^^ . a ; rdfs:label "relationship between Scaled Inverse Chi-Square and Scaled Inverse Chi-Square whereby \\\\text{If } X \\\\sim ScaledInverseChiSquare1\\\\nu,\\\\sigma \\\\text{ then } kX \\\\sim ScaledInverseChiSquare1\\\\nu, k\\\\sigma" ; ; ; "\\text{If } X \\sim ScaledInverseChiSquare1(\\nu,\\sigma) \\text{ then } kX \\sim ScaledInverseChiSquare1(\\nu, k\\sigma)"^^ ; "ScaledInverseChiSquare1(\\nu,\\sigma) \\rightarrow ScaledInverseChiSquare1(\\nu,\\sigma)"^^ ; "\\url{https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution}"^^ . a ; rdfs:label "variance of Dagum 1" ; "variance of Dagum 1"^^ ; """\\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}"""^^ . a ; rdfs:label "shape parameter of Kumaraswamy-1" ; "shape parameter of Kumaraswamy-1"^^ ; "Kumaraswamy1.shape2"^^ ; "b"^^ ; "b > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "relationship between Poisson 2 and Poisson 1 whereby \\\\lambda = \\\\exp\\\\alpha" ; ; ; "\\lambda = \\exp(\\alpha)"^^ ; "Poisson2(\\alpha) \\rightarrow Poisson1(\\lambda)"^^ ; "\\cite{stan-manual:2015b}"^^ . a ; rdfs:label "index parameter of Negative-Binomial-6" ; "index parameter of Negative-Binomial-6"^^ ; "NegativeBinomial6.dispersion"^^ ; "\\phi"^^ ; "\\phi \\in R, \\phi > 0"^^ ; "index parameter"^^ ; "dispersion"^^ . a ; rdfs:label "number of successes of Negative-Binomial-1" ; "number of successes of Negative-Binomial-1"^^ ; "NegativeBinomial1.numberOfSuccesses"^^ ; "r"^^ ; "r > 0, r \\in N"^^ ; "number of successes"^^ ; "numberOfSuccesses"^^ . a ; rdfs:label "location parameter of Johnson-SN-1" ; "location parameter of Johnson-SN-1"^^ ; "JohnsonSN1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location parameter"^^ ; "location"^^ . a ; rdfs:label "variance of Von Mises 1" ; "variance of Von Mises 1"^^ ; "1 - \\frac{I_1(\\kappa)}{I_1(0)} \\text{ (circular variance)}"^^ . a ; rdfs:label "median of Poisson 1" ; "median of Poisson 1"^^ ; "\\approx\\lfloor\\lambda+1/3-0.02/\\lambda\\rfloor"^^ . a ; rdfs:label "PMF of Binomial 1" ; "PMF of Binomial 1"^^ ; "{n \\choose k}\\, p^k (1-p)^{n-k}"^^ ; "choose(n,k) * p^k*(1-p)^(n-k)"^^ . a ; rdfs:label "PDF of Johnson SB 1" ; "PDF of Johnson SB 1"^^ ; "\\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)}"^^ ; "exp(-1/2*(gamma+delta*log((x-mu)/(-x+mu+sigma)))^2)*delta*sigma/(sqrt(2*pi)*(x-mu)*(-x+mu+sigma))"^^ . a ; rdfs:label "scale of Power-Normal-1" ; "scale of Power-Normal-1"^^ ; "PowerNormal1.scale"^^ ; "\\sigma"^^ ; "\\sigma \\in R, \\sigma > 0"^^ ; "scale "^^ ; "scale"^^ . a ; rdfs:label "Poisson intensity of Zero-inflated-Poisson-1" ; "Poisson intensity of Zero-inflated-Poisson-1"^^ ; "ZeroInflatedPoisson1.rate"^^ ; "\\lambda"^^ ; "\\lambda \\in R, \\lambda > 0"^^ ; "Poisson intensity"^^ ; "rate"^^ . a ; rdfs:label "HF of Uniform Discrete 2" ; "HF of Uniform Discrete 2"^^ ; "\\frac{1}{n-k}"^^ ; "1/(n-k)"^^ . a ; rdfs:label "degrees of freedom of Noncentral-F-1" ; "degrees of freedom of Noncentral-F-1"^^ ; "NoncentralF1.degreesOfFreedom1"^^ ; "m"^^ ; "m \\in N"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom1"^^ . a ; rdfs:label "variance of Inverse Chi-Square" ; "variance of Inverse Chi-Square"^^ ; "\\frac{2}{(\\nu - 2)^2(\\nu-4)} \\text{ for } \\nu>4"^^ . a ; rdfs:label "relationship between Negative Binomial 5 and Negative Binomial 1 whereby r = \\\\alpha, p = \\\\beta / 1 + \\\\beta" ; ; ; "r = \\alpha, p = \\beta / (1 + \\beta)"^^ ; "NegativeBinomial5(\\alpha, \\beta) \\rightarrow NegativeBinomial1(r,p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Negative Binomial 5" ; "CDF of Negative Binomial 5"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "variance of Log-Normal 1" ; "variance of Log-Normal 1"^^ ; "(e^{\\sigma^2}\\!\\!-1) e^{2\\mu+\\sigma^2}"^^ . a ; rdfs:label "CDF of Beta Negative Binomial1" ; "CDF of Beta Negative Binomial1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "mean of Negative Binomial 3" ; "mean of Negative Binomial 3"^^ ; "\\mu"^^ . a ; rdfs:label "Johnson SB 1" ; "Johnson SB 1"^^ ; "JohnsonSB1"^^ ; ; ; "x"^^ ; "x \\in (\\mu,\\mu+\\sigma"^^ ; , , , . a ; rdfs:label "relationship between Binomial 1 and Poisson 1 whereby \\\\lambda = np, n \\\\rightarrow \\\\infty " ; ; ; "\\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}"""^^ . a ; rdfs:label "log mean of Negative-Binomial-6" ; "log mean of Negative-Binomial-6"^^ ; "NegativeBinomial6.logMean"^^ ; "\\eta"^^ ; "\\eta \\in R"^^ ; "log mean"^^ ; "logMean"^^ . a ; rdfs:label "Multivariate Student T 2" ; "Multivariate (Student) T 2"^^ ; "MultivariateStudentT2"^^ ; ; "x"^^ ; "x \\in R^d, k\\geq 2"^^ ; , , . a ; rdfs:label "degrees of freedom of Multivariate-Student-T-1" ; "degrees of freedom of Multivariate-Student-T-1"^^ ; "MultivariateStudentT1.degreesOfFreedom"^^ ; "\\nu"^^ ; "\\nu \\ge 2"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "Inverse Binomial 1" ; "Inverse Binomial 1"^^ ; "InverseBinomial1"^^ ; ; ; ; "x"^^ ; "x \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "mode of Dagum 1" ; "mode of Dagum 1"^^ ; "b{\\left( \\frac{ap-1}{a+1} \\right)}^{\\tfrac{1}{a}}"^^ . a ; rdfs:label "relationship between Zero-Inflated Generalized Poisson 1 and Generalized Poisson 3 whereby p0=0" ; ; ; "p0=0"^^ ; "ZeroInflatedGeneralizedPoisson1(\\mu,\\alpha,p0) \\rightarrow GeneralizedPoisson3(\\mu,\\alpha)"^^ ; "\\cite{famoye2006zero}"^^ . a ; rdfs:label "Poisson intensity of Poisson-1" ; "Poisson intensity of Poisson-1"^^ ; "Poisson1.rate"^^ ; "\\lambda"^^ ; "\\lambda \\in R, \\lambda > 0"^^ ; "Poisson intensity"^^ ; "rate"^^ . a ; rdfs:label "CDF of Birnbaum-Saunders 1" ; "CDF of Birnbaum-Saunders 1"^^ ; """\\int_0^x f(x), \\text { with f the PDF}"""^^ ; "cumsum(PDF*rep(stepSize,length(PDF)))"^^ . a ; rdfs:label "variance of Zero-inflated Poisson 1" ; "variance of Zero-inflated Poisson 1"^^ ; "\\lambda (1-\\pi) (1 + \\lambda \\pi)"^^ . a ; rdfs:label """relationship between Hypergeometric 1 and Poisson 1 whereby X \\\\sim Hypergeometric1N, K, n \\\\Rightarrow Y\\\\sim Poisson1\\\\lambda \\\\text{ as K, N and n tend to infinity for } \\\\\\\\ K/N \\\\text{ small and } nK/N \\\\rightarrow \\\\lambda""" ; ; ; """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)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "mode of Von Mises 1" ; "mode of Von Mises 1"^^ ; "\\mu"^^ . a ; rdfs:label "inverse scale matrix of Wishart-2" ; "inverse scale matrix of Wishart-2"^^ ; "Wishart2.inverseScaleMatrix"^^ ; "R"^^ ; "p\\times p - \\text{symmetric, positive definite matrix}"^^ ; "inverse scale matrix"^^ ; "inverseScaleMatrix"^^ . a ; rdfs:label "scale parameter of Johnson-SN-1" ; "scale parameter of Johnson-SN-1"^^ ; "JohnsonSN1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label """relationship between Student's t-distribution 1 and Standard Normal 1 whereby X\\\\sim StudentT1\\\\nu \\\\Rightarrow StandardNormal10,1 \\\\text{ as } \\\\nu \\\\text{ tends to infinity.}\\\\\\\\ \\\\text{ The approximation is reasonable for } \\\\nu\\\\ge 30""" ; ; ; """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)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "noncentrality parameter of Noncentral-F-1" ; "noncentrality parameter of Noncentral-F-1"^^ ; "NoncentralF1.noncentrality"^^ ; "\\lambda"^^ ; "\\lambda \\geq 0"^^ ; "noncentrality parameter"^^ ; "noncentrality"^^ . a ; rdfs:label "variance of Uniform Discrete 2" ; "variance of Uniform Discrete 2"^^ ; "\\frac{n(n+2)}{12}"^^ . a ; rdfs:label "CDF of Power-Normal 1" ; "CDF of Power-Normal 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 }"""^^ . a ; rdfs:label "CDF of Inverse Chi-Square" ; "CDF of Inverse Chi-Square"^^ ; "\\Gamma\\Big(\\frac{\\nu}{2},\\frac{1}{2x}\\Big)\\Big/\\Gamma\\Big(\\frac{\\nu}{2}\\Big)"^^ ; "Igamma(nu/2,1/2/x, lower=F) / gamma(nu/2) "^^ . a ; rdfs:label "Negative Binomial 3" ; "Negative Binomial 3"^^ ; "NegativeBinomial3"^^ ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "PMF of Negative Binomial 5" ; "PMF of Negative Binomial 5"^^ ; "\\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"^^ . a ; rdfs:label "relationship between Weibull 1 and Rayleigh 1 whereby k=2, \\\\lambda=\\\\sqrt{2} \\\\sigma" ; ; ; "k=2, \\lambda=\\sqrt{2} \\sigma"^^ ; "Weibull1(\\lambda,k) \\rightarrow Rayleigh1(\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Negative Binomial 6" ; "variance of Negative Binomial 6"^^ ; "\\exp(\\eta) + \\exp(\\eta)^2 / \\phi"^^ . a ; rdfs:label "covariance matrix of Multivariate-Student-T-1" ; "covariance matrix of Multivariate-Student-T-1"^^ ; "MultivariateStudentT1.covarianceMatrix"^^ ; "\\Sigma"^^ ; "\\Sigma, \\text{ positive-definite real } p\\times p \\text{ matrix}"^^ ; "covariance matrix"^^ ; "covarianceMatrix"^^ . a ; rdfs:label "scale of Gompertz-1" ; "scale of Gompertz-1"^^ ; "Gompertz1.scale"^^ ; "b"^^ ; "b > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "median of Dagum 1" ; "median of Dagum 1"^^ ; "b{\\left(-1+2^{\\tfrac{1}{p}}\\right)}^{-\\tfrac{1}{a}}"^^ . a ; rdfs:label "Poisson 2" ; "Poisson 2"^^ ; "Poisson2"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; . a ; rdfs:label "number of elements of Zipf-1" ; "number of elements of Zipf-1"^^ ; "Zipf1.numberOfElements"^^ ; "n"^^ ; "n \\in N^+"^^ ; "number of elements"^^ ; "numberOfElements"^^ . a ; rdfs:label "CDF of Binomial 2" ; "CDF of Binomial 2"^^ ; "I_{1-logit^{-1}(\\alpha)}(n - k, 1 + k)"^^ ; """invLogit = function(x) { exp(x)/(1+exp(x)) } \\\\ Rbeta(1-invLogit(alpha), n-k, 1+k)"""^^ . a ; rdfs:label "PMF of Beta Negative Binomial1" ; "PMF of Beta Negative Binomial1"^^ ; "\\frac{{n-1+x \\choose x}{B(n+\\alpha,\\beta+x)}}{B(\\alpha,\\beta)}"^^ ; "choose(n-1+x,x)*beta(n+alpha,beta+x)/beta(alpha,beta)"^^ . a ; rdfs:label "median of Von Mises 1" ; "median of Von Mises 1"^^ ; "\\mu"^^ . a ; rdfs:label "mean of Zero-inflated Poisson 1" ; "mean of Zero-inflated Poisson 1"^^ ; "(1-\\pi) \\lambda"^^ . a ; rdfs:label "variance of Poisson 1" ; "variance of Poisson 1"^^ ; "\\lambda"^^ . a ; rdfs:label "variance of Noncentral F 1" ; "variance of Noncentral F 1"^^ ; "\\left(\\frac{n}{m}\\right)^2 \\frac{2}{(n-2)(n-4)}\\left[\\frac{(\\lambda+m)^2}{n-2} +2\\lambda + m \\right]"^^ . a ; rdfs:label "CDF of Dagum 1" ; "CDF of Dagum 1"^^ ; "\\left( 1+{\\left(\\frac{x}{b}\\right)}^{-a} \\right)^{-p}"^^ ; "(1+(x/b)^(-a))^(-p)"^^ . a ; rdfs:label "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)"^^ ; "StudentT1(\\nu) \\rightarrow F1(n_1,n_2)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TF.pdf}"""^^ . a ; rdfs:label "degrees of freedom of Wishart-2" ; "degrees of freedom of Wishart-2"^^ ; "Wishart2.degreesOfFreedom"^^ ; "k"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "Muth 1" ; "Muth 1"^^ ; "Muth1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; . a ; rdfs:label "mean of Uniform Discrete 2" ; "mean of Uniform Discrete 2"^^ ; "\\frac{n}{2}"^^ . a ; rdfs:label "transforming parameter of Power-Normal-1" ; "transforming parameter of Power-Normal-1"^^ ; "PowerNormal1.shape"^^ ; "\\lambda"^^ ; "\\lambda \\in R"^^ ; "transforming parameter"^^ ; "shape"^^ . a ; rdfs:label "relationship between Zero-Inflated Negative Binomial 1 and Negative Binomial 2 whereby p0=0" ; ; ; "p0=0"^^ ; "ZeroInflatedNegativeBinomial1(\\lambda,\\tau,p0) \\rightarrow NegativeBinomial2(\\lambda,\\tau)"^^ ; "\\cite{Troconiz:2009fv}"^^ . a ; rdfs:label "Negative Binomial 5" ; "Negative Binomial 5"^^ ; "NegativeBinomial5"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "mean of Inverse Chi-Square" ; "mean of Inverse Chi-Square"^^ ; "\\frac{1}{\\nu - 2} \\text{ for } \\nu>2"^^ . a ; rdfs:label "shape of Gompertz-1" ; "shape of Gompertz-1"^^ ; "Gompertz1.shape"^^ ; "\\eta"^^ ; "\\eta > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "mean of Dagum 1" ; "mean of Dagum 1"^^ ; """\\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}"""^^ . a ; rdfs:label "relationship between Log-Normal 1 and Log-Normal 5 whereby \\\\mu = \\\\mu, \\\\tau = 1 / \\\\sigma^2" ; ; ; "\\mu = \\mu, \\tau = 1 / \\sigma^2"^^ ; "LogNormal1(\\mu,\\sigma) \\rightarrow LogNormal5(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "location of Multivariate-Student-T-1" ; "location of Multivariate-Student-T-1"^^ ; "MultivariateStudentT1.mean"^^ ; "\\mu"^^ ; "\\mu = [\\mu_1, \\dots, \\mu_p]^T, \\mu_i \\in R"^^ ; "location"^^ ; "mean"^^ . a ; rdfs:label "mean of Negative Binomial 6" ; "mean of Negative Binomial 6"^^ ; "\\exp(\\eta)"^^ . a ; rdfs:label "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) "^^ ; "\\cite{shmueli2005useful}"^^ . a ; rdfs:label "PMF of Negative Binomial 3" ; "PMF of Negative Binomial 3"^^ ; "\\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"^^ . a ; rdfs:label "shape of Zipf-1" ; "shape of Zipf-1"^^ ; "Zipf1.shape"^^ ; "\\alpha"^^ ; "\\alpha \\in R, \\alpha > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "shape of Log-Normal-1" ; "shape of Log-Normal-1"^^ ; "LogNormal1.stdevLog"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "shape"^^ ; "stdevLog"^^ . a ; rdfs:label "Beta Negative Binomial1" , "Beta Pascal"^^ ; "Beta Negative Binomial1"^^ ; "BetaNegativeBinomial1"^^ ; ; ; ; "x"^^ ; "x \\in \\{0,\\dots,n\\}"^^ ; , , . a ; rdfs:label "number of trials of Binomial-2" ; "number of trials of Binomial-2"^^ ; "Binomial2.numberOfTrials"^^ ; "n"^^ ; "n \\in N, n \\ge 0"^^ ; "number of trials"^^ ; "numberOfTrials"^^ . a ; rdfs:label "median of Log-Normal 4" ; "median of Log-Normal 4"^^ ; "m"^^ . a ; rdfs:label "Sinh-Arcsinh 2" , "SHASH"^^ ; "Sinh-Arcsinh 2"^^ ; "SinhArcsinh2"^^ ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , , , . a ; rdfs:label "mean of Normal-1" ; "mean of Normal-1"^^ ; "Normal1.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "degrees of freedom of NoncentralT" ; "degrees of freedom of NoncentralT"^^ ; "NoncentralT1.degreesOfFreedom"^^ ; "n"^^ ; "n \\in N"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "relationship between Generalized Gamma 2 and Weibull 1 whereby c=1, a=0, b=\\\\lambda" ; ; ; "c=1, a=0, b=\\lambda"^^ ; "GeneralizedGamma2(a,b,c,k) \\rightarrow Weibull1(\\lambda,k)"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "variance of Multivariate Student T 1" ; "variance of Multivariate (Student) T 1"^^ ; """\\begin{cases} \\frac{\\nu}{\\nu-2} \\Sigma & \\text{for }\\nu > 2 \\\\ undefined & \\text{else} \\end{cases}"""^^ . a ; rdfs:label "PMF of Double Poisson 1" ; "PMF of Double Poisson 1"^^ ; """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)"""^^ ; """K(mu,phi) * phi^(1/2) * exp(-phi*mu) * (exp(-y)*y^y)/factorial(y) * (exp(1)*mu/y)^(phi*y) \\\\ # with K(mu,phi) = 1 / (1 + (1-phi)/(12*mu*phi)*(1 + 1/(mu*phi)) )"""^^ . a ; rdfs:label "minimum of Log-Uniform-1" ; "minimum of Log-Uniform-1"^^ ; "LogUniform1.minimum"^^ ; "min"^^ ; "min>0"^^ ; "minimum"^^ ; "minimum"^^ . a ; rdfs:label "Inverted-chi-square"^^ , "Inverse Chi-Square" ; "Inverse Chi-Square"^^ ; "InverseChiSquare1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; . a ; rdfs:label "mean of Log-Normal 1" ; "mean of Log-Normal 1"^^ ; "e^{\\mu+\\sigma^2/2}"^^ . a ; rdfs:label "location parameter of Asymmetric-Laplace-1" ; "location parameter of Asymmetric-Laplace-1"^^ ; "AsymmetricLaplace1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location parameter"^^ ; "location"^^ . a ; rdfs:label "relationship between Binomial 2 and Binomial 1 whereby p = \\\\exp\\\\alpha / 1 + \\\\exp\\\\alpha" ; ; ; "p = \\exp(\\alpha) / (1 + \\exp(\\alpha))"^^ ; "Binomial2(n,\\alpha) \\rightarrow Binomial1(n,p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Erlang 1" ; "variance of Erlang 1"^^ ; "b^2 c"^^ . a ; rdfs:label "HF of Burr 1" ; "HF of Burr 1"^^ ; "\\frac{\\frac{kc}{\\alpha}(x/\\alpha)^{c-1}}{1+(x/\\alpha)^c} "^^ ; "(k*c)/alpha * (x/alpha)^(c-1) / (1+(x/alpha)^c)"^^ . a ; rdfs:label """relationship between Log-Normal 5 and Log-Normal 2 whereby \\\\mu = \\\\mu, v = 1 / \\\\tau""" ; ; ; """\\mu = \\mu, v = 1 / \\tau"""^^ ; "LogNormal5(\\mu,\\tau) \\rightarrow LogNormal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "median / geometric mean of Log-Normal-3" ; "median / geometric mean of Log-Normal-3"^^ ; "LogNormal3.median"^^ ; "m"^^ ; "m>0"^^ ; "median / geometric mean"^^ ; "median"^^ . a ; rdfs:label "relationship between Amoroso 1 and Chi-squared 1 whereby \\\\text{ChiSquared1}k \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a = 0, \\\\theta=2, \\\\alpha=k/2, \\\\beta=1" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "SF of Log-Logistic 2" ; "SF of Log-Logistic 2"^^ ; "\\frac{1}{1+(\\lambda x)^\\kappa}"^^ ; "1/(1+(lambda*x)^kappa)"^^ . a ; rdfs:label "degrees of freedom of Chi-squared-1" ; "degrees of freedom of Chi-squared-1"^^ ; "ChiSquared1.degreesOfFreedom"^^ ; "k"^^ ; "k \\in N"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "scale parameter of Dagum-1" ; "scale parameter of Dagum-1"^^ ; "Dagum1.scale"^^ ; "b"^^ ; "b > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "standard deviation of Normal-1" ; "standard deviation of Normal-1"^^ ; "Normal1.stdev"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "standard deviation"^^ ; "stdev"^^ . a ; rdfs:label "PMF of Binomial 2" ; "PMF of Binomial 2"^^ ; "{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)"""^^ . a ; rdfs:label "mean of Log-Normal 4" ; "mean of Log-Normal 4"^^ ; "m \\sqrt{cv^2 + 1}"^^ . a ; rdfs:label "noncentrality parameter of NoncentralT" ; "noncentrality parameter of NoncentralT"^^ ; "NoncentralT1.noncentrality"^^ ; "\\lambda"^^ ; "\\lambda \\geq 0"^^ ; "noncentrality parameter"^^ ; "noncentrality"^^ . a ; rdfs:label "maximum of Log-Uniform-1" ; "maximum of Log-Uniform-1"^^ ; "LogUniform1.maximum"^^ ; "max"^^ ; "max \\geq min"^^ ; "maximum"^^ ; "maximum"^^ . a ; rdfs:label "mode of Multivariate Student T 1" ; "mode of Multivariate (Student) T 1"^^ ; "\\mu"^^ . a ; rdfs:label "CDF of Double Poisson 1" ; "CDF of Double Poisson 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "PDF of Inverse Chi-Square" ; "PDF of Inverse Chi-Square"^^ ; "\\frac{2^{-\\nu/2}}{\\Gamma(\\nu/2)} x^{-(\\nu/2+1)} \\exp\\Big(-\\frac{1}{2x}\\Big) "^^ ; "2^(-nu/2)/gamma(nu/2)*x^(-(nu/2+1))*exp(-1/(2*x))"^^ . a ; rdfs:label "CDF of Log-Normal 1" ; "CDF of Log-Normal 1"^^ ; "\\frac12 + \\frac12\\,\\text{erf}\\Big[\\frac{\\log x-\\mu}{\\sqrt{2}\\sigma}\\Big]"^^ ; "1/2 + 1/2 *erf( (log(x)-mu)/(sqrt(2)*sigma) )"^^ . a ; rdfs:label "CDF of Burr 1" ; "CDF of Burr 1"^^ ; "1 - \\frac{1}{\\Big[1+(x/\\alpha)^c\\Big]^k}"^^ ; "1 - 1/(1+(x/alpha)^c)^k"^^ . a ; rdfs:label "relationship between Generalized Poisson 1 and Poisson 1 whereby \\\\delta = 0, \\\\theta=\\\\mu" ; ; ; "\\delta = 0, \\theta=\\mu"^^ ; "GeneralizedPoisson1(\\theta,\\delta) \\rightarrow Poisson1(\\lambda)"^^ ; "\\cite{Yang:2007vn}"^^ . a ; rdfs:label "variance of Chi-squared 1" ; "variance of Chi-squared 1"^^ ; "2k"^^ . a ; rdfs:label "scale parameter of Asymmetric-Laplace-1" ; "scale parameter of Asymmetric-Laplace-1"^^ ; "AsymmetricLaplace1.scale"^^ ; "\\sigma"^^ ; "\\sigma \\geq 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "SF of Gamma 1" ; "SF of Gamma 1"^^ ; "1 - \\frac{\\Gamma(\\beta,x/\\alpha)}{\\Gamma(\\beta)}"^^ . a ; rdfs:label "Level start of Trapezoidal-1" ; "Level start of Trapezoidal-1"^^ ; "Trapezoidal1.levelStart"^^ ; "b"^^ ; "a \\leq b < c"^^ ; "Level start"^^ ; "levelStart"^^ . a ; rdfs:label "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)"^^ ; "\\cite{LavielleBook:2014}"^^ . a ; rdfs:label "variance of Log-Normal 3" ; "variance of Log-Normal 3"^^ ; "m^2 e^{\\sigma^2} [e^{\\sigma^2}-1]"^^ . a ; rdfs:label "mean of NoncentralT" ; "mean of NoncentralT"^^ ; "sqrt(n/2)*gamma((n-1)/2) / gamma(n/2)*delta"^^ . a owl:DatatypeProperty ; rdfs:label "parameter has mathematical type" . a ; rdfs:label """relationship between Log-Normal 3 and Log-Normal 5 whereby \\\\mu = \\\\logm, \\\\tau = 1 / \\\\sigma^2""" ; ; ; """\\mu = \\log(m), \\tau = 1 / \\sigma^2"""^^ ; "LogNormal3(m,\\sigma) \\rightarrow LogNormal5(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mode of Log-Normal 1" ; "mode of Log-Normal 1"^^ ; "e^{\\mu-\\sigma^2}"^^ . a ; rdfs:label "shape parameter of Dagum-1" ; "shape parameter of Dagum-1"^^ ; "Dagum1.shape2"^^ ; "a"^^ ; "a > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "variance of Log-Normal 4" ; "variance of Log-Normal 4"^^ ; "m^2 (cv^2+1) cv^2"^^ . a ; rdfs:label "Upper bound of Trapezoidal-1" ; "Upper bound of Trapezoidal-1"^^ ; "Trapezoidal1.upperBound"^^ ; "d"^^ ; "c \\leq d"^^ ; "Upper bound"^^ ; "upperBound"^^ . a ; rdfs:label "mode of Normal 1" ; "mode of Normal 1"^^ ; "\\mu"^^ . a ; rdfs:label "mean of Log-Uniform 1" ; "mean of Log-Uniform 1"^^ ; "\\frac{max-min}{\\log(max) - \\log(min)}"^^ . a ; rdfs:label "median of Multivariate Student T 1" ; "median of Multivariate (Student) T 1"^^ ; "\\mu"^^ . a ; rdfs:label "mean of Erlang 1" ; "mean of Erlang 1"^^ ; "bc"^^ . a ; rdfs:label "PDF of Burr 1" ; "PDF of Burr 1"^^ ; "\\frac{\\frac{kc}{\\alpha}(x/\\alpha)^{c-1}}{\\Big[1+(x/\\alpha)^c\\Big]^{k+1}}"^^ ; "k*c/alpha * (x/alpha)^(c-1) / (1+(x/alpha)^c)^(k+1)"^^ . a ; rdfs:label "mean of Asymmetric Laplace 1" ; "mean of Asymmetric Laplace 1"^^ ; "\\mu"^^ . a ; rdfs:label "location of Laplace-2" ; "location of Laplace-2"^^ ; "Laplace2.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "shape of Weibull-1" ; "shape of Weibull-1"^^ ; "Weibull1.shape"^^ ; "k"^^ ; "k\\in (0, +\\infty)"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "CDF of Binomial 1" ; "CDF of Binomial 1"^^ ; "I_{1-p}(n - k, 1 + k)"^^ ; "Rbeta(1-p, n-k, 1+k)"^^ . a ; rdfs:label "Level end of Trapezoidal-1" ; "Level end of Trapezoidal-1"^^ ; "Trapezoidal1.levelEnd"^^ ; "c"^^ ; "b < c \\leq d"^^ ; "Level end"^^ ; "levelEnd"^^ . a ; rdfs:label "PDF of Logistic 1" ; "PDF of Logistic 1"^^ ; "\\frac{e^{-\\frac{x-\\mu}{s}}} {s\\left(1+e^{-\\frac{x-\\mu}{s}}\\right)^2}"^^ ; "exp(-(x-mu)/s) / (s*(1+exp(-(x-mu)/s))^2)"^^ . a owl:Class ; rdfs:label "Limiting" ; rdfs:subClassOf . a ; rdfs:label "mode of Log-Normal 3" ; "mode of Log-Normal 3"^^ ; "m/e^{\\sigma^2}"^^ . a ; rdfs:label "relationship between Amoroso 1 and Chi 1 whereby \\\\text{Chi1}k \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a = 0, \\\\theta=\\\\sqrt{2}, \\\\alpha=k/2, \\\\beta=2" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "standard deviation of Standard-Normal-1" ; "standard deviation of Standard-Normal-1"^^ ; "StandardNormal1.stdev"^^ ; "\\sigma"^^ ; "\\sigma=1"^^ ; "standard deviation"^^ ; "stdev"^^ . a ; rdfs:label "PDF of NoncentralT" ; "PDF of NoncentralT"^^ ; "\\frac{e^{-\\frac{\\delta^2}{2}}}{\\sqrt{n\\pi} \\;\\Gamma\\left(\\frac{n}2\\right)} \\sum^{\\infty}_{r=0} \\frac{(t' \\delta)^r}{r! n^{\\frac{r}2}} \\left(1+\\frac{t'^2}{n}\\right)^{-\\frac{n+r+1}{2}}2^{\\frac{r}{2}} \\, \\Gamma\\left(\\frac{n+r+1}{2}\\right)"^^ ; """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) }"""^^ . a ; rdfs:label "relationship between Weibull 1 and Weibull 2 whereby v=k, \\\\lambda = 1/\\\\lambda^k" ; ; ; "v=k, \\lambda = 1/\\lambda^k"^^ ; "Weibull1(\\lambda,k) \\rightarrow Weibull2(\\lambda,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mode of Log-Normal 4" ; "mode of Log-Normal 4"^^ ; "m/(cv^2+1)"^^ . a ; rdfs:label "shape parameter of Dagum-1" ; "shape parameter of Dagum-1"^^ ; "Dagum1.shape1"^^ ; "p"^^ ; "p > 0"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "median of Log-Normal 1" ; "median of Log-Normal 1"^^ ; "e^{\\mu}"^^ . a ; rdfs:label "variance of Normal 1" ; "variance of Normal 1"^^ ; "\\sigma^2"^^ . a ; rdfs:label "mean of Multivariate Student T 1" ; "mean of Multivariate (Student) T 1"^^ ; """\\begin{cases} \\mu & \\text{for }\\nu > 1 \\\\ undefined & \\text{else} \\end{cases}"""^^ . a ; rdfs:label "Double Poisson 1" ; "Double Poisson 1"^^ ; "DoublePoisson1"^^ ; ; ; ; ; "x"^^ ; "x \\in \\{1,2,3,\\dots,n\\}"^^ ; , . a ; rdfs:label "variance of Log-Uniform 1" ; "variance of Log-Uniform 1"^^ ; """\\frac{max^2-min^2}{2[\\log(max) - \\log(min)]} - \\Big(\\frac{max-min}{\\log(max) - \\log(min)}\\Big)^2"""^^ . a ; rdfs:label "mode of Erlang 1" ; "mode of Erlang 1"^^ ; "b(c-1), c \\ge 1"^^ . a ; rdfs:label "Singh-Maddala"^^ , "Burr 1" , "Burr Type XII"^^ ; "Burr 1"^^ ; "Burr1"^^ ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , , . a ; rdfs:label "variance of Asymmetric Laplace 1" ; "variance of Asymmetric Laplace 1"^^ ; "\\mu^2 + 2\\sigma^2"^^ . a ; rdfs:label "Logistic 1" ; "Logistic 1"^^ ; "Logistic1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a owl:DatatypeProperty ; rdfs:label "has alternate name" . a ; rdfs:label "precision of Laplace-2" ; "precision of Laplace-2"^^ ; "Laplace2.tau"^^ ; "\\tau"^^ ; "\\tau > 0, \\tau \\in R"^^ ; "precision"^^ ; "tau"^^ . a owl:DatatypeProperty ; rdfs:label "has Latex expression of support" . a ; rdfs:label "median of Log-Normal 3" ; "median of Log-Normal 3"^^ ; "m"^^ . a ; rdfs:label "relationship between Binomial 1 and Bernoulli 1 whereby n=1" ; ; ; "n=1"^^ ; "Binomial1(n,p) \\rightarrow Bernoulli1(p)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BinomialBernoulli.pdf}"""^^ . a ; rdfs:label "mean of Arcsine 1" ; "mean of Arcsine 1"^^ ; "\\frac{1}{2}"^^ . a ; rdfs:label "CDF of Multivariate Student T 1" ; "CDF of Multivariate (Student) T 1"^^ ; "\\text{no analytic expression}"^^ . a ; rdfs:label "Generalized Gamma 1" ; "Generalized Gamma 1"^^ ; "GeneralizedGamma1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , , . a ; rdfs:label "Poisson intensity of Double-Poisson-1" ; "Poisson intensity of Double-Poisson-1"^^ ; "DoublePoisson1.rate"^^ ; "\\mu"^^ ; "\\mu \\in R, \\mu > 0"^^ ; "Poisson intensity"^^ ; "rate"^^ . a ; rdfs:label """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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape of Log-Normal-2" ; "shape of Log-Normal-2"^^ ; "LogNormal2.varLog"^^ ; "v"^^ ; "v > 0"^^ ; "shape"^^ ; "varLog"^^ . a owl:Ontology . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Standard Uniform 1" ; "CDF of Standard Uniform 1"^^ ; "x"^^ ; "x"^^ . a owl:DatatypeProperty ; rdfs:label "has variate of mathematical type" . a ; rdfs:label "NoncentralT" ; "NoncentralT"^^ ; "NoncentralT1"^^ ; ; ; "t'=\\frac{x}{\\sqrt{y/n}} \\text{ with } x \\sim Normal2(\\delta,1), y \\sim ChiSquared1(n)."^^ ; "t' \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "variance of Zero-Inflated Generalized Poisson 1" ; "variance of Zero-Inflated Generalized Poisson 1"^^ ; "(1-p0)[\\mu^2 + \\mu(1 + \\alpha \\mu)^2] - (1-p0)^2 \\mu^2"^^ . a ; rdfs:label "mean of Log-Normal 3" ; "mean of Log-Normal 3"^^ ; "m e^{\\frac{1}{2}\\sigma^2}"^^ . a ; rdfs:label "shape of Two-Sided-Power-1" ; "shape of Two-Sided-Power-1"^^ ; "TwoSidedPower1.shape"^^ ; "n"^^ ; "n > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "CDF of Log-Normal 5" ; "CDF of Log-Normal 5"^^ ; "\\frac12 + \\frac12\\,\\text{erf}\\Big[\\frac{\\log x-\\mu}{\\sqrt{2/\\tau}}\\Big]"^^ ; "1/2 + 1/2 * erf( (log(x)-mu) / sqrt(2/tau) )"^^ . a ; rdfs:label "mean of Chi-squared 1" ; "mean of Chi-squared 1"^^ ; "k"^^ . a ; rdfs:label "CDF of Bernoulli 1" ; "CDF of Bernoulli 1"^^ ; """\\begin{cases} 0 & \\text{for }k<0 \\\\ q & \\text{for }0\\leq k<1 \\\\ 1 & \\text{for }k\\geq 1 \\end{cases}"""^^ . a ; rdfs:label "coefficient of variation of Log-Normal-4" ; "coefficient of variation of Log-Normal-4"^^ ; "LogNormal4.coefVar"^^ ; "cv"^^ ; "cv>0"^^ ; "coefficient of variation"^^ ; "coefVar"^^ . a ; rdfs:label "variance of Benford 1" ; "variance of Benford 1"^^ ; "\\approx 6.0565"^^ . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution} "^^ . a ; rdfs:label "mean of Laplace 2" ; "mean of Laplace 2"^^ ; "\\mu"^^ . a ; rdfs:label "HF of Erlang 1" ; "HF of Erlang 1"^^ ; "\\frac{(x/b)^{c-1}}{b(c-1)! \\sum^{c-1}_{i=0} \\frac{(x/b)^i}{i!}}"^^ . a ; rdfs:label "lower triangular matrix of Multivariate-Normal-3" ; "lower triangular matrix of Multivariate-Normal-3"^^ ; "MultivariateNormal3.choleskyFactor"^^ ; "L"^^ ; "L \\in R^{K \\times K}, \\text{ lower triangular and such that } LL^T \\text{ is positive definite}, \\text{with } K \\in N"^^ ; "lower triangular matrix"^^ ; "choleskyFactor"^^ . a ; rdfs:label "PDF of Asymmetric Laplace 1" ; "PDF of Asymmetric Laplace 1"^^ ; """\\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) }"""^^ . a ; rdfs:label "median of Arcsine 1" ; "median of Arcsine 1"^^ ; "\\frac{1}{2}"^^ . a ; rdfs:label "dispersion of Double-Poisson-1" ; "dispersion of Double-Poisson-1"^^ ; "DoublePoisson1.dispersion"^^ ; "\\phi"^^ ; "\\phi \\in R"^^ ; "dispersion"^^ ; "dispersion"^^ . a ; rdfs:label "degrees of freedom of Student-s-t-distribution-1" ; "degrees of freedom of Student-s-t-distribution-1"^^ ; "StudentT1.degreesOfFreedom"^^ ; "\\nu"^^ ; "\\nu > 0, \\nu \\in R"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "PDF of Multivariate Student T 1" ; "PDF of Multivariate (Student) T 1"^^ ; "\\frac{\\Gamma\\left[(\\nu+p)/2\\right]}{\\Gamma(\\nu/2)\\nu^{p/2}\\pi^{p/2}\\left|\\Sigma\\right|^{1/2}\\left[1+\\frac{1}{\\nu}(x-\\mu)^{\\rm T}\\Sigma^{-1}(x-\\mu)\\right]^{(\\nu+p)/2}}"^^ . a ; rdfs:label "mean of logx of Log-Normal-2" ; "mean of log(x) of Log-Normal-2"^^ ; "LogNormal2.meanLog"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean of log(x)"^^ ; "meanLog"^^ . a owl:DatatypeProperty ; rdfs:label "parameter has mathematical type" . a ; rdfs:label "mean of Zero-Inflated-Generalized-Poisson-1" ; "mean of Zero-Inflated-Generalized-Poisson-1"^^ ; "ZeroInflatedGeneralizedPoisson1.mean"^^ ; "\\mu"^^ ; "\\mu > 0"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "CDF of Half-normal 2" ; "CDF of Half-normal 2"^^ ; "erf\\Big(\\frac{x-\\mu}{\\sqrt{2}\\sigma}\\Big)"^^ ; "erf((x-mu) / (sqrt(2)*sigma))"^^ . a ; rdfs:label "CDF of Chi-squared 1" ; "CDF of Chi-squared 1"^^ ; "\\frac{1}{\\Gamma\\left(\\frac{k}{2}\\right)}\\; \\gamma\\left(\\frac{k}{2},\\,\\frac{x}{2}\\right)"^^ ; "1/gamma(k/2) * Igamma(k/2,x/2)"^^ . a ; rdfs:label "PDF of Log-Normal 5" ; "PDF of Log-Normal 5"^^ ; "\\sqrt{\\frac{\\tau}{2 \\pi}} \\frac{1}{x}e^{-\\frac{\\tau}{2}(\\log x-\\mu)^2}"^^ ; "sqrt(tau / (2*pi)) * (1/x) * exp(- (tau/2)*(log(x)-mu)^2 )"^^ . a ; rdfs:label "CDF of Log-Normal 3" ; "CDF of Log-Normal 3"^^ ; "\\frac12 + \\frac12\\,\\text{erf}\\Big[\\frac{\\log x-\\log m}{\\sqrt{2}\\sigma}\\Big]"^^ ; "1/2 + 1/2 * erf( (log(x)-log(m)) / (sqrt(2)*sigma) )"^^ . a ; rdfs:label "location of Two-Sided-Power-1" ; "location of Two-Sided-Power-1"^^ ; "TwoSidedPower1.location"^^ ; "m"^^ ; "a < m < b"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "variance of Laplace 2" ; "variance of Laplace 2"^^ ; "2 / \\tau^2"^^ . a ; rdfs:label "relationship between Weibull 2 and Weibull 1 whereby k=v, \\\\lambda = \\\\lambda^{-1/v}" ; ; ; "k=v, \\lambda = \\lambda^{-1/v}"^^ ; "Weibull2(\\lambda,v) \\rightarrow Weibull1(\\lambda,k)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mode of Benford 1" ; "mode of Benford 1"^^ ; "1"^^ . a ; rdfs:label "median / geometric mean of Log-Normal-4" ; "median / geometric mean of Log-Normal-4"^^ ; "LogNormal4.median"^^ ; "m"^^ ; "m>0"^^ ; "median / geometric mean"^^ ; "median"^^ . a ; rdfs:label "CDF of Asymmetric Laplace 1" ; "CDF of Asymmetric Laplace 1"^^ ; """\\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)}"""^^ . a ; rdfs:label "location of Multivariate-Normal-3" ; "location of Multivariate-Normal-3"^^ ; "MultivariateNormal3.mean"^^ ; "\\mu"^^ ; "\\mu \\in R^K"^^ ; "location"^^ ; "mean"^^ . a ; rdfs:label "SF of Erlang 1" ; "SF of Erlang 1"^^ ; "\\Big[\\exp\\Big(-\\frac{x}{b}\\Big)\\Big]\\Big( \\sum^{c-1}_{i=0} \\frac{(x/b)^i}{i!}\\Big)"^^ . a ; rdfs:label "Multivariate Student T 1" ; "Multivariate (Student) T 1"^^ ; "MultivariateStudentT1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in R^p"^^ ; , , . a ; rdfs:label "mode of Categorical Ordered 1" ; "mode of Categorical Ordered 1"^^ ; "i\\text{ such that }p_i=\\max(p_1, \\ldots, p_k)"^^ . a ; rdfs:label "CDF of Generalized Gamma 1" ; "CDF of Generalized Gamma 1"^^ ; "\\frac{\\gamma(d/p, (x/a)^p)}{\\Gamma(d/p)}"^^ ; "Igamma(d/p, (x/a)^p, lower=T) / gamma(d/p)"^^ . a ; rdfs:label "mean of Double Poisson 1" ; "mean of Double Poisson 1"^^ ; "\\mu"^^ . a ; rdfs:label "Standard Uniform 1" , "Rectangular"^^ ; "Standard Uniform 1"^^ ; "StandardUniform1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,1]"^^ ; , . a ; rdfs:label "PMF of Zero-Inflated Generalized Poisson 1" ; "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}"""^^ ; """PMF1=p0 + (1-p0) * exp(-mu/(1+ alpha*mu)) # for y = 0 PMF2=(1-p0)*( mu/(1+alpha*mu))^y*((1+alpha*y)^(y-1))/factorial(y)*exp((-mu*(1+alpha*y))/(1+alpha*mu)) # for y > 0"""^^ . a owl:DatatypeProperty ; rdfs:label "has variate symbol in Latex" . a ; rdfs:label "Categorical Nonordered 1" ; "Categorical Nonordered 1"^^ ; "CategoricalNonordered1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in \\{1,\\dots,k\\}"^^ ; . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/Rice_distribution}"^^ . a ; rdfs:label "relationship between Zero-Inflated Generalized Poisson 1 and Poisson 1 whereby p0=0, \\\\alpha=0, \\\\lambda=\\\\mu" ; ; ; "p0=0, \\alpha=0, \\lambda=\\mu"^^ ; "ZeroInflatedGeneralizedPoisson1(\\mu,\\alpha,p0) \\rightarrow Poisson1(\\lambda)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "median of Benford 1" ; "median of Benford 1"^^ ; "3"^^ . a ; rdfs:label "mode of Chi-squared 1" ; "mode of Chi-squared 1"^^ ; "max\\{k-2,0\\}"^^ . a ; rdfs:label "PDF of Laplace 2" ; "PDF of Laplace 2"^^ ; "\\frac{\\tau}{2} \\exp \\left(-\\tau|x-\\mu| \\right)"^^ ; "tau/2 * exp(-tau * abs(x-mu))"^^ . a ; rdfs:label "SF of Weibull 2" ; "SF of Weibull 2"^^ ; "\\exp(-x^v \\lambda)"^^ ; "exp(-x^v * lambda)"^^ . a ; rdfs:label "mean of Normal 2" ; "mean of Normal 2"^^ ; "\\mu"^^ . a ; rdfs:label "scale parameter of Burr-1" ; "scale parameter of Burr-1"^^ ; "Burr1.scale"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "scale of Half-normal-2" ; "scale of Half-normal-2"^^ ; "HalfNormal2.scale"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "PDF of Erlang 1" ; "PDF of Erlang 1"^^ ; "\\frac{(x/b)^{c-1} \\exp(-x/b)}{b(c-1)!}"^^ ; "((x/b)^(c-1) *exp(-x/b)) / (b * factorial(c-1))"^^ . a ; rdfs:label "shape parameter of Weibull-Discrete-1" ; "shape parameter of Weibull-Discrete-1"^^ ; "WeibullDiscrete1.shape1"^^ ; "p"^^ ; "0 ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "variance of Multivariate Normal 3" ; "variance of Multivariate Normal 3"^^ ; "LL^T"^^ . a ; rdfs:label "variance of Double Poisson 1" ; "variance of Double Poisson 1"^^ ; "\\mu/\\phi"^^ . a ; rdfs:label "median of Categorical Ordered 1" ; "median of Categorical Ordered 1"^^ ; "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"^^ . a ; rdfs:label "PDF of Generalized Gamma 1" ; "PDF of Generalized Gamma 1"^^ ; "\\frac{p/a^d}{\\Gamma(d/p)} x^{d-1}e^{-(x/a)^p}"^^ ; "p/a^d/gamma(d/p) * x^(d-1) * exp(-(x/a)^p)"^^ . a ; rdfs:label "SF of Arcsine 1" ; "SF of Arcsine 1"^^ ; "\\frac{\\pi -2 \\arcsin\\left(2x -1\\right)}{2\\pi}"^^ ; "(pi - 2 * asin(2*x -1)) / (2*pi)"^^ . a owl:ObjectProperty ; rdfs:label "has variate of mathematical type" . a ; rdfs:label "CDF of Zero-Inflated Generalized Poisson 1" ; "CDF of Zero-Inflated Generalized Poisson 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "c(PMF1,cumsum(PMF2)+PMF1)"^^ . a ; rdfs:label "mean of Log-Normal 5" ; "mean of Log-Normal 5"^^ ; "e^{\\mu + \\frac{1}{2\\tau}}"^^ . a ; rdfs:label "PDF of Standard Uniform 1" ; "PDF of Standard Uniform 1"^^ ; "1"^^ ; "1"^^ . a ; rdfs:label "PMF of Categorical Nonordered 1" ; "PMF of Categorical Nonordered 1"^^ ; "p(x=i)=p_i"^^ . a ; rdfs:label "relationship between Log-Normal 1 and Normal 1 whereby \\\\logX" ; ; ; "\\log(X)"^^ ; "LogNormal1(\\mu,\\sigma) \\rightarrow Normal1(\\mu,\\sigma)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalLognormal.pdf}"""^^ . a ; rdfs:label "Scaled Inverse Chi-Square" ; "Scaled Inverse Chi-Square"^^ ; "ScaledInverseChiSquare1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "relationship between Exponential 2 and F 1 whereby \\\\text{If } X_1, X_2 \\\\sim Exponential21 \\\\text{ mutually independent and identically distributed} \\\\\\\\ \\\\text{ random variables } \\\\Rightarrow X_1/X_2 \\\\text{ has the } F1 \\\\text{ distribution}" ; ; ; "\\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}"^^ ; "Exponential2(1) \\rightarrow F1(n_1,n_2)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ExponentialF.pdf}"""^^ . a ; rdfs:label "median of Normal 2" ; "median of Normal 2"^^ ; "\\mu"^^ . a ; rdfs:label "HF of Log-Logistic 2" ; "HF of Log-Logistic 2"^^ ; "\\frac{\\lambda \\kappa (\\lambda x)^{\\kappa-1}}{1+(\\lambda x)^\\kappa}"^^ ; "lambda * kappa * (lambda * x)^(kappa-1) / (1+(lambda * x)^kappa)"^^ . a ; rdfs:label "mean of Zero-Inflated Generalized Poisson 1" ; "mean of Zero-Inflated Generalized Poisson 1"^^ ; "(1-p0)\\mu"^^ . a ; rdfs:label "CDF of Laplace 2" ; "CDF of Laplace 2"^^ ; """\\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}"""^^ ; """1/2 * exp( tau*(x-mu) ) # for x < mu 1- 1/2 * exp( -tau*(x-mu) ) # x >= mu"""^^ . a ; rdfs:label "median of Chi-squared 1" ; "median of Chi-squared 1"^^ ; "\\approx k\\bigg(1-\\frac{2}{9k}\\bigg)^3"^^ . a ; rdfs:label "SF of Burr 1" ; "SF of Burr 1"^^ ; "\\frac{1}{\\Big[1+(x/\\alpha)^c\\Big]^k}"^^ ; "1/(1+(x/alpha)^c)^k"^^ . a ; rdfs:label "Asymmetric Laplace 1" ; "Asymmetric Laplace 1"^^ ; "AsymmetricLaplace1"^^ ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "CDF of Zero-inflated Poisson 1" ; "CDF of Zero-inflated Poisson 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "c(PMF1,cumsum(PMF2)+PMF1)"^^ . a ; rdfs:label "SF of Weibull Discrete 1" ; "SF of Weibull Discrete 1"^^ ; "(1-p)^{x^\\beta}"^^ ; "(1-p)^(x^beta)"^^ . a ; rdfs:label "location of Half-normal-2" ; "location of Half-normal-2"^^ ; "HalfNormal2.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "CDF of Erlang 1" ; "CDF of Erlang 1"^^ ; "1 - \\Big[\\exp\\Big(-\\frac{x}{b}\\Big)\\Big]\\Big( \\sum^{c-1}_{i=0} \\frac{(x/b)^i}{i!}\\Big)"^^ ; "R function"^^ . a ; rdfs:label "relationship between Beta 1 and Arcsine 1 whereby \\\\alpha=1/2, \\\\beta = 1/2" ; ; ; "\\alpha=1/2, \\beta = 1/2"^^ ; "Beta1(\\alpha,\\beta) \\rightarrow Arcsine1"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/UDR.html}"^^ . a ; rdfs:label "median of Conway-Maxwell-Poisson 1" ; "median of Conway-Maxwell-Poisson 1"^^ ; "\\text{No closed form}"^^ . a ; rdfs:label "mean of Half-normal 2" ; "mean of Half-normal 2"^^ ; "\\mu + \\sigma \\sqrt{\\frac{2}{\\pi}}"^^ . a owl:Class ; rdfs:label "non denumerable set" ; rdfs:subClassOf . a ; rdfs:label "shape parameter of Weibull-Discrete-1" ; "shape parameter of Weibull-Discrete-1"^^ ; "WeibullDiscrete1.shape2"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "Lomax 1" ; "Lomax 1"^^ ; "Lomax1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; , . a ; rdfs:label "relationship between Two-Sided Power 1 and Uniform 1 whereby n=1" ; ; ; "n=1"^^ ; "TwoSidedPower1(a,b,m,n) \\rightarrow Uniform1(a,b)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TSPUniform.pdf}"^^ . a ; rdfs:label "variance of Frechet 1" ; "variance of Frechet 1"^^ ; """\\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}"""^^ . a owl:DatatypeProperty ; rdfs:label "distribution relationship definition in Latex" . a ; rdfs:label "mode of Cauchy 1" ; "mode of Cauchy 1"^^ ; "x_0"^^ . a ; rdfs:label "scale of Exponential-2" ; "scale of Exponential-2"^^ ; "Exponential2.mean"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "scale"^^ ; "mean"^^ . a ; rdfs:label "minimum of Uniform-1" ; "minimum of Uniform-1"^^ ; "Uniform1.minimum"^^ ; "a"^^ ; "a \\in R"^^ ; "minimum"^^ ; "minimum"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "median of Levy 1" ; "median of Levy 1"^^ ; "c/2(\\textrm{erfc}^{-1}(1/2))^2 \\text{ for } \\mu=0"^^ . a owl:DatatypeProperty ; rdfs:label "has variate symbol" . a ; rdfs:label "scale of Pareto-Type-I" ; "scale of Pareto-Type-I"^^ ; "ParetoTypeI1.scale"^^ ; "x_m"^^ ; "x_m > 0, x_m \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "CDF of Standard Normal 1" ; "CDF of Standard Normal 1"^^ ; "\\frac12\\left[1 + \\text{erf}\\left( \\frac{x}{\\sqrt{2}}\\right)\\right]"^^ ; "1/2 * (1 + erf(x/(sqrt(2))))"^^ . a ; rdfs:label "relationship between Amoroso 1 and Rayleigh 1 whereby \\\\text{Rayleigh1}\\\\sigma \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a = 0, \\\\theta=\\\\sqrt{2\\\\sigma^2}, \\\\alpha=1, \\\\beta=2" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "PDF of Multivariate Gaussian Process Distribution 1" ; "PDF of Multivariate Gaussian Process Distribution 1"^^ ; "\\prod^K_{i=1} \\text{MultivariateNormal1}(x_i|0,w^{-1}_i \\Sigma)"^^ . a ; rdfs:label "relationship between Bernoulli 2 and Bernoulli 1 whereby p = \\\\exp\\\\alpha / 1 + \\\\exp\\\\alpha" ; ; ; "p = \\exp(\\alpha) / (1 + \\exp(\\alpha))"^^ ; "Bernoulli2(\\alpha) \\rightarrow Bernoulli1(p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Generalized Gamma 1" ; "mean of Generalized Gamma 1"^^ ; "a \\frac{\\Gamma((d+1)/p)}{\\Gamma(d/p)}"^^ . a ; rdfs:label "mode of Log-Normal 7" ; "mode of Log-Normal 7"^^ ; "\\frac{\\mu_N}{(1+\\sigma_N^2/\\mu_N^2)^{3/2}}"^^ . a ; rdfs:label "mean of Categorical Nonordered 1" ; "mean of Categorical Nonordered 1"^^ ; "E([x=i]) = p_i, \\text{this is the mean of the Iverson bracket } [x=i] \\text{ and not the mean of } x"^^ . a owl:DatatypeProperty ; rdfs:label "has Latex symbol" . a ; rdfs:label "relationship between Pareto Type II and Lomax 1 whereby \\\\mu=0" ; ; ; "\\mu=0"^^ ; "ParetoTypeII1(\\mu,\\lambda,\\alpha) \\rightarrow Lomax1(\\lambda,\\alpha)"^^ ; "\\url{https://en.wikipedia.org/wiki/Pareto_distribution#Pareto_types_I.E2.80.93IV}"^^ . a ; rdfs:label "relationship between Amoroso 1 and Normal 1 whereby \\\\text{Normal1}\\\\mu,\\\\sigma \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a=\\\\mu-\\\\sigma\\\\sqrt{\\\\alpha}, \\\\theta=\\\\sigma/\\\\sqrt{\\\\alpha}, \\\\beta=1 \\\\text{ and } \\\\alpha \\\\rightarrow \\\\infty" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "variance of Half-normal 2" ; "variance of Half-normal 2"^^ ; "\\sigma^2 (1-\\frac{2}{\\pi})"^^ . a ; rdfs:label "variance of Conway-Maxwell-Poisson 1" ; "variance of Conway-Maxwell-Poisson 1"^^ ; "\\sum_{j=0}^\\infty \\frac{j^2\\lambda^j}{(j!)^\\nu Z(\\lambda, \\nu)} - \\text{Mean}^2"^^ . a ; rdfs:label "median of Cauchy 1" ; "median of Cauchy 1"^^ ; "x_0"^^ . a owl:ObjectProperty ; rdfs:label "distribution relationship to" . a ; rdfs:label "PDF of Lomax 1" ; "PDF of Lomax 1"^^ ; " {\\alpha \\over \\lambda} \\left[{1+ {x \\over \\lambda}}\\right]^{-(\\alpha+1)}"^^ ; "alpha/lambda* (1+ x/lambda)^(-(alpha+1))"^^ . a ; rdfs:label "Borel 1" ; "Borel 1"^^ ; "Borel1"^^ ; ; ; "k"^^ ; "k \\in \\{1,2,\\dots,\\infty\\}"^^ ; . a owl:Class ; rdfs:label "denumerable set" ; rdfs:subClassOf . a ; rdfs:label "shape of Frechet-1" ; "shape of Frechet-1"^^ ; "Frechet1.shape"^^ ; "\\alpha"^^ ; "\\alpha \\in R^+"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "relationship between Two-Sided Power 1 and Triangular 1 whereby n=2" ; ; ; "n=2"^^ ; "TwoSidedPower1(a,b,m,n) \\rightarrow Triangular1(a,b,m)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/TSPTriangular.pdf}"^^ . a ; rdfs:label "Log-Normal 2" , "lognormal"^^ , "Galton"^^ ; "Log-Normal 2"^^ ; "LogNormal2"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a owl:DatatypeProperty ; rdfs:label "has type name" . a ; rdfs:label "variance of Pareto Type I" ; "variance of Pareto Type I"^^ ; """\\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}"""^^ . a ; rdfs:label "PDF of Standard Normal 1" ; "PDF of Standard Normal 1"^^ ; "\\frac{e^{-\\frac{1}{2} x^2}}{\\sqrt{2\\pi}}"^^ ; "1/(sqrt(2*pi))*exp(-x^2/2)"^^ . a ; rdfs:label "relationship between Sinh-Arcsinh 2 and Normal 1 whereby \\\\epsilon=0,\\\\delta=1" ; ; ; "\\epsilon=0,\\delta=1"^^ ; "SinhArcsinh2(\\mu,\\sigma,\\epsilon,\\delta) \\rightarrow Normal1(\\mu,\\sigma)"^^ ; "\\cite{rubio2013modelling}"^^ . a ; rdfs:label "mode of Levy 1" ; "mode of Levy 1"^^ ; "\\frac{c}{3} \\text{ for } \\mu=0"^^ . a ; rdfs:label "PMF of Borel 1" ; "PMF of Borel 1"^^ ; "\\frac{e^{\\mu n} (\\mu n)^{n-1}}{n!}"^^ ; "e^(mu*n)*(mu*n)^(n-1) / factorial(n)"^^ . a ; rdfs:label "Multivariate Gaussian Process Distribution 1" ; "Multivariate Gaussian Process Distribution 1"^^ ; "MultivariateGaussianProcess1"^^ ; ; "x"^^ ; "x \\in R^{K\\times N}"^^ ; , . a ; rdfs:label "relationship between Truncated Normal 1 and Half-normal 1 whereby \\\\mu=0, a=0, b=\\\\infty" ; ; ; "\\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}"""^^ . a owl:DatatypeProperty ; rdfs:label "has comment" . a ; rdfs:label "relationship between Negative Binomial 3 and Negative Binomial 1 whereby r=\\\\phi, p=r/\\\\mu+r" ; ; ; "r=\\phi, p=r/(\\mu+r)"^^ ; "NegativeBinomial3(\\mu, \\phi) \\rightarrow NegativeBinomial1(r, p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Categorical Nonordered 1" ; "CDF of Categorical Nonordered 1"^^ ; "undefined"^^ . a ; rdfs:label "mode of Generalized Gamma 1" ; "mode of Generalized Gamma 1"^^ ; "a \\left(\\frac{d-1}{p}\\right)^{\\frac{1}{p}}, \\text{ for } d>1"^^ . a ; rdfs:label "median of Log-Normal 7" ; "median of Log-Normal 7"^^ ; "\\frac{\\mu_N}{\\sqrt{1+\\sigma_N^2/\\mu_N^2}}"^^ . a ; rdfs:label "CDF of Conway-Maxwell-Poisson 1" ; "CDF of Conway-Maxwell-Poisson 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "Log-Normal 3" , "lognormal"^^ , "Galton"^^ ; "Log-Normal 3"^^ ; "LogNormal3"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "relationship between Arcsine 2 and Arcsine 1 whereby a=0, b=1" ; ; ; "a=0, b=1"^^ ; "Arcsine2(a,b) \\rightarrow Arcsine1"^^ ; "ProbOnto spec"^^ . a owl:ObjectProperty ; rdfs:label "distribution relationship from" . a owl:Class ; rdfs:label "enumeration" ; rdfs:subClassOf . a ; rdfs:label "PDF of Half-normal 2" ; "PDF of Half-normal 2"^^ ; "\\sqrt{\\frac{2}{\\pi}} \\frac{1}{\\sigma} \\exp\\Big[-\\frac{1}{2}\\Big(\\frac{x-\\mu}{\\sigma}\\Big)^2\\Big]"^^ ; "sqrt(2/pi)*1/sigma * exp(-1/2 * ((x-mu)/sigma)^2)"^^ . a ; rdfs:label """relationship between Log-Normal 2 and Log-Normal 6 whereby m=\\\\exp\\\\mu, \\\\sigma_g=\\\\exp\\\\sqrt{v}""" ; ; ; """m=\\exp(\\mu), \\sigma_g=\\exp(\\sqrt{v})"""^^ ; "LogNormal2(\\mu,v) \\rightarrow LogNormal6(m,\\sigma_g)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Cauchy 1" ; "mean of Cauchy 1"^^ ; "undefined"^^ . a ; rdfs:label "scale of Frechet-1" ; "scale of Frechet-1"^^ ; "Frechet1.scale"^^ ; "\\sigma"^^ ; "\\sigma \\in R^+"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "relationship between Standard Two-Sided Power 1 and Standard Uniform 1 whereby n=1" ; ; ; "n=1"^^ ; "StandardTwoSidedPower1(m,n) \\rightarrow StandardUniform1"^^ . a ; rdfs:label "relationship between Amoroso 1 and Inverse Chi-Square whereby \\\\text{InverseChiSquare1}k \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } a = 0, \\\\theta=1/2, \\\\alpha=k/2, \\\\beta=-1" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "variance of Levy 1" ; "variance of Levy 1"^^ ; "\\infty"^^ . a ; rdfs:label "Generalized Poisson 1" ; "Generalized Poisson 1"^^ ; "GeneralizedPoisson1"^^ ; ; ; ; ; """k """^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a owl:DatatypeProperty ; rdfs:label "has short name" . a ; rdfs:label "mode of Exponential 2" ; "mode of Exponential 2"^^ ; "0"^^ . a ; rdfs:label "relationship between Log-Normal 6 and Log-Normal 7 whereby \\\\mu_N = m \\\\exp\\\\big\\\\frac12 \\\\log^2\\\\sigma_g\\\\big, \\\\sigma_N = m \\\\exp\\\\big\\\\frac12 \\\\log^2\\\\sigma_g\\\\big \\\\sqrt{\\\\exp\\\\big[\\\\log^2\\\\sigma_g\\\\big]-1 }" ; ; ; "\\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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Log-Normal 7" ; "mean of Log-Normal 7"^^ ; "\\mu_N"^^ . a ; rdfs:label "symmetric, positive definite kernel matrix of Multivariate-Gaussian-Process-Distribution-1" ; "symmetric, positive definite kernel matrix of Multivariate-Gaussian-Process-Distribution-1"^^ ; "MultivariateGaussianProcess1.kernelMatrix"^^ ; "\\Sigma"^^ ; "\\Sigma \\in R^{N \\times N}, N \\in N"^^ ; "symmetric, positive definite kernel matrix"^^ ; "kernelMatrix"^^ . a ; rdfs:label "variance of Generalized Gamma 1" ; "variance of Generalized Gamma 1"^^ ; "a^2\\left(\\frac{\\Gamma((d+2)/p)}{\\Gamma(d/p)} - \\left(\\frac{\\Gamma((d+1)/p)}{\\Gamma(d/p)}\\right)^2\\right)"^^ . a ; rdfs:label "location of Generalized-Gamma-2" ; "location of Generalized-Gamma-2"^^ ; "GeneralizedGamma2.location"^^ ; "a"^^ ; "a > 0"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "median of Standard Normal 1" ; "median of Standard Normal 1"^^ ; "0"^^ . a ; rdfs:label "mode of Pareto Type I" ; "mode of Pareto Type I"^^ ; "x_m"^^ . a ; rdfs:label "CDF of Lomax 1" ; "CDF of Lomax 1"^^ ; "1- \\left[{1+ {x \\over \\lambda}}\\right]^{-\\alpha}"^^ ; "1- (1+ x/lambda)^(-alpha)"^^ . a ; rdfs:label "CDF of Standard Triangular 1" ; "CDF of Standard Triangular 1"^^ ; """\\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}"""^^ ; """CDF1=1/2*x^2+x+1/2 CDF2=-1/2*x^2+x+1/2"""^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "CDF of Borel 1" ; "CDF of Borel 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a owl:DatatypeProperty ; rdfs:label "has MathML symbol" . a ; rdfs:label "Standard Triangular 1" ; "Standard Triangular 1"^^ ; "StandardTriangular1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "0 \\leq x \\leq 1"^^ . a ; rdfs:label "PDF of Log-Normal 3" ; "PDF of Log-Normal 3"^^ ; "\\frac{1}{x\\sigma\\sqrt{2\\pi}}\\ e^{-\\frac{\\left[\\log (x/m)\\right]^2}{2\\sigma^2}}"^^ ; "1/(x*sigma*sqrt(2*pi)) * exp(-(log(x/m))^2 / (2*sigma^2))"^^ . a ; rdfs:label "mean of Conway-Maxwell-Poisson 1" ; "mean of Conway-Maxwell-Poisson 1"^^ ; "\\sum_{j=0}^\\infty \\frac{j\\lambda^j}{(j!)^\\nu Z(\\lambda, \\nu)}"^^ . a ; rdfs:label "CDF of Cauchy 1" ; "CDF of Cauchy 1"^^ ; "\\frac{1}{\\pi} \\arctan\\left(\\frac{x-x_0}{\\gamma}\\right)+\\frac{1}{2}"^^ ; "1/pi * atan((x-x0)/gamma)+1/2"^^ . a owl:Class ; rdfs:label "Model formulation" ; rdfs:subClassOf . a owl:Class ; rdfs:label "infinite set" ; rdfs:subClassOf . a ; rdfs:label "relationship between Amoroso 1 and Half-normal 2 whereby \\\\text{HalfNormal2}\\\\mu,\\\\sigma \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution for } \\\\mu=0 \\\\text{ and when } a = 0, \\\\theta=\\\\sqrt{2\\\\sigma^2}, \\\\alpha=1/2, \\\\beta=2" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "relationship between Log-Normal 3 and Log-Normal 1 whereby \\\\mu = \\\\logm, \\\\sigma = \\\\sigma" ; ; ; "\\mu = \\log(m), \\sigma = \\sigma"^^ ; "LogNormal3(m,\\sigma) \\rightarrow LogNormal1(\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape parameter of Noncentral-Beta-1" ; "shape parameter of Noncentral-Beta-1"^^ ; "NoncentralBeta1.shape2"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "lognormal"^^ , "Log-Normal 5" , "Galton"^^ ; "Log-Normal 5"^^ ; "LogNormal5"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "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)"^^ ; "\\url{https://en.wikipedia.org/wiki/L%C3%A9vy_distribution#Related_distributions}"^^ . a ; rdfs:label "relationship between Sinh-Arcsinh 2 and Standard Normal 1 whereby \\\\mu=0, \\\\sigma=1, \\\\epsilon=0,\\\\delta=1" ; ; ; "\\mu=0, \\sigma=1, \\epsilon=0,\\delta=1"^^ ; "SinhArcsinh2(\\mu,\\sigma,\\epsilon,\\delta) \\rightarrow StandardNormal1"^^ ; "\\cite{rubio2013modelling}"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Exponential 2" ; "variance of Exponential 2"^^ ; "\\beta^2"^^ . a ; rdfs:label "maximum of Uniform-1" ; "maximum of Uniform-1"^^ ; "Uniform1.maximum"^^ ; "b"^^ ; "b \\in R, a < b"^^ ; "maximum"^^ ; "maximum"^^ . a ; rdfs:label "scale of Generalized-Gamma-1" ; "scale of Generalized-Gamma-1"^^ ; "GeneralizedGamma1.scale"^^ ; "a"^^ ; "a > 0"^^ ; "scale"^^ ; "scale"^^ . a owl:DatatypeProperty ; rdfs:label "has plain text symbol" . a ; rdfs:label "scale of Generalized-Gamma-2" ; "scale of Generalized-Gamma-2"^^ ; "GeneralizedGamma2.scale"^^ ; "b"^^ ; "b > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "relationship between Standard Uniform 1 and Beta 1 whereby aX_1, X_2,... , X_n iid \\\\sim StandardNormal1 \\\\Rightarrow \\\\text{ with } \\\\alpha=r \\\\text{ and } \\\\beta=n-r+1, X_{r} \\\\sim Beta\\\\alpha,\\\\beta" ; ; ; "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}"""^^ . a owl:DatatypeProperty ; rdfs:label "parameter has definition expressed in MathML" . a ; rdfs:label "rate of Borel-1" ; "rate of Borel-1"^^ ; "Borel1.shape"^^ ; "\\mu"^^ ; "\\mu \\in (0,1)"^^ ; "rate"^^ ; "shape"^^ . a ; rdfs:label "PDF of Standard Triangular 1" ; "PDF of Standard Triangular 1"^^ ; """\\begin{cases} x+1 & \\text{ for } -1 < x < 0 \\\\ 1-x & \\text{ for } 0 \\leq x < 1 \\end{cases}"""^^ ; """x+1 for -1 < x < 0 \\\\ 1-x for 0 <= x < 1 """^^ . a ; rdfs:label "mean of Standard Normal 1" ; "mean of Standard Normal 1"^^ ; "0"^^ . a ; rdfs:label "mean of Lomax 1" ; "mean of Lomax 1"^^ ; "\\frac{\\lambda}{\\alpha -1} \\text{ for } \\alpha > 1 \\text { otherwise undefined }"^^ . a ; rdfs:label "median of Pareto Type I" ; "median of Pareto Type I"^^ ; "x_m \\sqrt[\\alpha]{2}"^^ . a ; rdfs:label "shape parameter of Noncentral-Beta-1" ; "shape parameter of Noncentral-Beta-1"^^ ; "NoncentralBeta1.shape1"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "shape parameter"^^ ; "shape1"^^ . a owl:Class ; rdfs:label "Special case, Reparameterisation" ; rdfs:subClassOf . a ; rdfs:label "standard deviation of Exponentially-modified-Gaussian-1" ; "standard deviation of Exponentially-modified-Gaussian-1"^^ ; "ExponentiallyModifiedGaussian1.stdev"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "standard deviation"^^ ; "stdev"^^ . a ; rdfs:label "relationship between Scaled Inverse Chi-Square and Inverse Chi-Square whereby \\\\mbox{If } X \\\\sim \\\\mbox{Scale-inv-}\\\\chi^2\\\\nu, \\\\tau^2 \\\\mbox{ then } \\\\frac{X}{\\\\tau^2 \\\\nu} \\\\sim \\\\mbox{inv-}\\\\chi^2\\\\nu" ; ; ; "\\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"^^ ; "\\url{https://en.wikipedia.org/wiki/Scaled_inverse_chi-squared_distribution}"^^ . a ; rdfs:label "location of Skew-Normal" ; "location of Skew-Normal"^^ ; "SkewNormal1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "PDF of Half-normal 1" ; "PDF of Half-normal 1"^^ ; "\\frac{2\\theta}{\\pi} e^{-\\theta^2 x^2 / \\pi}"^^ ; "2*theta/pi * exp(-theta^2 * x^2 / pi)"^^ . a ; rdfs:label "shape of Generalized-Gamma-1" ; "shape of Generalized-Gamma-1"^^ ; "GeneralizedGamma1.shape1"^^ ; "d"^^ ; "d > 0"^^ ; "shape"^^ ; "shape1"^^ . a ; rdfs:label "scale of Gumbel-1" ; "scale of Gumbel-1"^^ ; "Gumbel1.scale"^^ ; "\\beta"^^ ; "\\beta>0, \\beta \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "relationship between Normal 1 and Chi-squared 1 whereby \\\\text{If } X_i \\\\sim N\\\\mu,\\\\sigma, i=1,2,...,n \\\\text{ are mutually independent and identically}\\\\\\\\\\\\text{ distributed random variables and } Y=\\\\sum_{i=1}^n X_i - \\\\mu/\\\\sigma^2 \\\\Rightarrow Y \\\\sim ChiSquared1n" ; ; ; "\\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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/NormalChisquare.pdf}"^^ . a ; rdfs:label "PDF of Cauchy 1" ; "PDF of Cauchy 1"^^ ; "\\frac{1}{\\pi\\gamma\\,\\left[1 + \\left(\\frac{x-x_0}{\\gamma}\\right)^2\\right]}"^^ ; "1 / (pi*gamma*(1 + ((x-x0)^2/gamma^2)))"^^ . a ; rdfs:label "mean of Gamma 2" ; "mean of Gamma 2"^^ ; "r / \\mu"^^ . a ; rdfs:label "relationship between Bernoulli 1 and Bernoulli 2 whereby \\\\alpha = \\\\logp/1-p" ; ; ; "\\alpha = \\log(p/(1-p))"^^ ; "Bernoulli1(p) \\rightarrow Bernoulli2(\\alpha)"^^ ; "ProbOnto spec"^^ . a owl:DatatypeProperty ; rdfs:label "has plain text expression" . a ; rdfs:label "Half Cauchy 1" ; "Half Cauchy 1"^^ ; "HalfCauchy1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ . a ; rdfs:label "category probabilities of Categorical-Nonordered-1" ; "category probabilities of Categorical-Nonordered-1"^^ ; "CategoricalNonordered1.categoryProb"^^ ; "p_1, \\ldots, p_k"^^ ; "0 \\leq p_i \\leq 1, \\Sigma p_i = 1"^^ ; "category probabilities"^^ ; "categoryProb"^^ . a ; rdfs:label "mean of Log-Normal 2" ; "mean of Log-Normal 2"^^ ; "e^{\\mu+v/2}"^^ . a ; rdfs:label "mode of Standard Uniform 1" ; "mode of Standard Uniform 1"^^ ; "\\text{any value in }[0,1]"^^ . a ; rdfs:label "shape parameter of Johnson-SL-1" ; "shape parameter of Johnson-SL-1"^^ ; "JohnsonSL1.shape2"^^ ; "\\delta"^^ ; "\\delta > 0"^^ ; "shape parameter"^^ ; "shape2"^^ . a ; rdfs:label "LKJ Correlation 1" ; "LKJ Correlation 1"^^ ; "LKJCorrelation1"^^ ; ; "\\Sigma"^^ ; "\\text{positive-definite, symmetric matrix with unit diagonal (i.e., a correlation matrix)}"^^ ; . a ; rdfs:label "median / geometric mean of Log-Normal-6" ; "median / geometric mean of Log-Normal-6"^^ ; "LogNormal6.median"^^ ; "m"^^ ; "m>0"^^ ; "median / geometric mean"^^ ; "median"^^ . a ; rdfs:label "CDF of Generalized Poisson 1" ; "CDF of Generalized Poisson 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "CDF of Logistic 1" ; "CDF of Logistic 1"^^ ; "\\frac{1}{1+e^{-\\frac{x-\\mu}{s}}}"^^ ; "1/(1+exp(-(x-mu)/s))"^^ . a ; rdfs:label "noncentrality parameter of Noncentral-Beta-1" ; "noncentrality parameter of Noncentral-Beta-1"^^ ; "NoncentralBeta1.noncentrality"^^ ; "\\lambda"^^ ; "\\lambda \\geq 0"^^ ; "noncentrality parameter"^^ ; "noncentrality"^^ . a ; rdfs:label "Half-normal 2" ; "Half-normal 2"^^ ; "HalfNormal2"^^ ; ; ; ; ; "x"^^ ; "x \\in [\\mu,+\\infty)"^^ ; , . a ; rdfs:label "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)"^^ ; """\\cite{crooks2010amoroso} """^^ . a ; rdfs:label "Half-normal 1" ; "Half-normal 1"^^ ; "HalfNormal1"^^ ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a owl:Class ; rdfs:label "Reparameterisation, Limiting" ; rdfs:subClassOf . a ; rdfs:label "relationship between Negative Binomial 3 and Negative Binomial 2 whereby \\\\lambda=\\\\mu, \\\\tau=1/\\\\phi" ; ; ; "\\lambda=\\mu, \\tau=1/\\phi"^^ ; "NegativeBinomial3(\\mu, \\phi) \\rightarrow NegativeBinomial2(\\lambda, \\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Gamma 2" ; "variance of Gamma 2"^^ ; "r / \\mu^2"^^ . a ; rdfs:label "Nakagami 1" ; "Nakagami 1"^^ ; "Nakagami1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "shape of Generalized-Gamma-1" ; "shape of Generalized-Gamma-1"^^ ; "GeneralizedGamma1.shape2"^^ ; "p"^^ ; "p > 0"^^ ; "shape"^^ ; "shape2"^^ . a ; rdfs:label "mean of Frechet 1" ; "mean of Frechet 1"^^ ; """\\begin{cases} \\ \\sigma\\Gamma\\left(1-\\frac{1}{\\alpha}\\right) & \\text{for } \\alpha>1 \\\\ \\ \\infty & \\text{otherwise} \\end{cases}"""^^ . a ; rdfs:label "rate or inverse scale of Exponentially-modified-Gaussian-1" ; "rate or inverse scale of Exponentially-modified-Gaussian-1"^^ ; "ExponentiallyModifiedGaussian1.rate"^^ ; "\\lambda"^^ ; "\\lambda > 0"^^ ; "rate or inverse scale"^^ ; "rate"^^ . a ; rdfs:label "Breit-Wigner"^^ , "Cauchy 1" , "Lorentz"^^ ; "Cauchy 1"^^ ; "Cauchy1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "variance of Categorical Nonordered 1" ; "variance of Categorical Nonordered 1"^^ ; "Var([x=i]) = p_i (1-p_i); \\quad Cov([x=i],[x=j]) = - p_i p_j~~(i\\neq j)"^^ . a owl:DatatypeProperty ; rdfs:label "has Latex expression" . a ; rdfs:label "median of Standard Uniform 1" ; "median of Standard Uniform 1"^^ ; "0.5"^^ . a ; rdfs:label "median of Log-Normal 2" ; "median of Log-Normal 2"^^ ; "e^{\\mu}"^^ . a ; rdfs:label "PDF of Half Cauchy 1" ; "PDF of Half Cauchy 1"^^ ; "\\frac2\\pi \\frac1{1+x^2}"^^ ; "2/pi * 1 / (1 + x^2)"^^ . a ; rdfs:label "shape parameter of Johnson-SL-1" ; "shape parameter of Johnson-SL-1"^^ ; "JohnsonSL1.shape1"^^ ; "\\gamma"^^ ; "\\gamma \\in R"^^ ; "shape parameter"^^ ; "shape1"^^ . a ; rdfs:label "lognormal"^^ , "Log-Normal 4" , "Galton"^^ ; "Log-Normal 4"^^ ; "LogNormal4"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , . a ; rdfs:label "PDF of LKJ Correlation 1" ; "PDF of LKJ Correlation 1"^^ ; "\\propto \\det(\\Sigma)^{(\\eta-1)}"^^ . a ; rdfs:label "PMF of Generalized Poisson 1" ; "PMF of Generalized Poisson 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)"^^ . a ; rdfs:label "shape of Beta-1" ; "shape of Beta-1"^^ ; "Beta1.beta"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape"^^ ; "beta"^^ . a owl:DatatypeProperty ; rdfs:label "distribution relationship reference" . a ; rdfs:label "shape of Skew-Normal" ; "shape of Skew-Normal"^^ ; "SkewNormal1.shape"^^ ; "\\alpha"^^ ; "\\alpha \\in R"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "median of Frechet 1" ; "median of Frechet 1"^^ ; "\\frac{\\sigma}{\\sqrt[\\alpha]{\\log_e(2)}}"^^ . a ; rdfs:label "variance of Exponentially modified Gaussian 1" ; "variance of Exponentially modified Gaussian 1"^^ ; "\\sigma^2 + 1/\\lambda^2"^^ . a ; rdfs:label "CDF of Noncentral Beta 1" ; "CDF of Noncentral Beta 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) }"""^^ . a ; rdfs:label "variance of Generalized Poisson 1" ; "variance of Generalized Poisson 1"^^ ; "\\frac{\\theta}{(1 -\\delta)^3}"^^ . a ; rdfs:label "variance of Normal-2" ; "variance of Normal-2"^^ ; "Normal2.var"^^ ; "v"^^ ; "v>0"^^ ; "variance"^^ ; "var"^^ . a ; rdfs:label "shape of Gamma-2" ; "shape of Gamma-2"^^ ; "Gamma2.shape"^^ ; "r"^^ ; "r > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "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)"^^ ; "ProbOnto spec"^^ . a owl:ObjectProperty ; rdfs:label "distribution has PDF" . a ; rdfs:label "Poisson intensity of Conway-Maxwell-Poisson-1" ; "Poisson intensity of Conway-Maxwell-Poisson-1"^^ ; "ConwayMaxwellPoisson1.rate"^^ ; "\\lambda"^^ ; "\\lambda \\in R, \\lambda > 0"^^ ; "Poisson intensity"^^ ; "rate"^^ . a ; rdfs:label "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)"^^ ; "Chi1(k) \\rightarrow ChiSquared1(k)"^^ ; "\\url{https://en.wikipedia.org/wiki/Chi_distribution}"^^ . a ; rdfs:label "CDF of Half Cauchy 1" ; "CDF of Half Cauchy 1"^^ ; "\\frac2\\pi \\arctan(x)"^^ ; "2/pi * atan(x)"^^ . a ; rdfs:label "mode of Dirichlet 1" ; "mode of Dirichlet 1"^^ ; "x_i = \\frac{\\alpha_i - 1}{\\sum_{i=1}^K\\alpha_i - K}, \\quad \\alpha_i > 1"^^ . a ; rdfs:label "mean of Standard Uniform 1" ; "mean of Standard Uniform 1"^^ ; "0.5"^^ . a ; rdfs:label "mode of Categorical Nonordered 1" ; "mode of Categorical Nonordered 1"^^ ; "i\\text{ such that }p_i=\\max(p_1, \\ldots, p_k)"^^ . a ; rdfs:label "Standard Normal 1" , "Standard Gaussian"^^ ; "Standard Normal 1"^^ ; "StandardNormal1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in R"^^ ; , . a ; rdfs:label "shape of LKJ-Correlation-1" ; "shape of LKJ-Correlation-1"^^ ; "LKJCorrelation1.shape"^^ ; "\\eta"^^ ; "\\eta > 0"^^ ; "shape "^^ ; "shape"^^ . a ; rdfs:label "PDF of Log-Normal 4" ; "PDF of Log-Normal 4"^^ ; "\\frac{1}{x\\sqrt{\\log(cv^2+1)}\\sqrt{2\\pi}}\\ e^{-\\frac{\\left[\\log (x/m)\\right]^2}{2\\ln(cv^2+1)}}"^^ ; "1/(x*sqrt(log(cv^2+1))*sqrt(2*pi)) * exp( -(log(x/m))^2 / (2*log(cv^2+1)) )"^^ . a ; rdfs:label "PDF of Log-Normal 2" ; "PDF of Log-Normal 2"^^ ; "\\frac{1}{x\\sqrt{v}\\sqrt{2\\pi}}\\ e^{-\\frac{\\left(\\log x-\\mu\\right)^2}{2 v}}"^^ ; "1/(x*sqrt(v)*sqrt(2*pi)) * exp(-(ln(x)-mu)^2/(2*v))"^^ . a owl:DatatypeProperty ; rdfs:label "distribution relationship pair expression Latex" . a ; rdfs:label "location of Pareto-Type-II" ; "location of Pareto-Type-II"^^ ; "ParetoTypeII1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "scale of Skew-Normal" ; "scale of Skew-Normal"^^ ; "SkewNormal1.scale"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "relationship between Amoroso 1 and Levy 1 whereby \\\\text{Levy1}\\\\mu,c \\\\text{ distribution is a special case of the Amoroso1} a,\\\\theta,\\\\alpha,\\\\beta \\\\text{ distribution when } \\\\theta=c/2, \\\\alpha=1/2, \\\\beta=-1" ; ; ; "\\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)"^^ ; "\\cite{crooks2010amoroso}"^^ . a ; rdfs:label "shape of Beta-1" ; "shape of Beta-1"^^ ; "Beta1.alpha"^^ ; "\\alpha"^^ ; "\\alpha > 0"^^ ; "shape"^^ ; "alpha"^^ . a ; rdfs:label "PDF of Noncentral Beta 1" ; "PDF of Noncentral Beta 1"^^ ; "\\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) }"""^^ . a ; rdfs:label "mean of Exponentially-modified-Gaussian-1" ; "mean of Exponentially-modified-Gaussian-1"^^ ; "ExponentiallyModifiedGaussian1.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "mode of Frechet 1" ; "mode of Frechet 1"^^ ; "\\sigma\\left(\\frac{\\alpha}{1+\\alpha}\\right)^{1/\\alpha}"^^ . a ; rdfs:label "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}"""^^ . a ; rdfs:label "mean of Normal-2" ; "mean of Normal-2"^^ ; "Normal2.mean"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "rate of Gamma-2" ; "rate of Gamma-2"^^ ; "Gamma2.rate"^^ ; "\\mu"^^ ; "\\mu > 0"^^ ; "rate"^^ ; "rate"^^ . a ; rdfs:label "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)"^^ ; """\\url{https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution} \\\\ \\cite{stan-manual:2015b}"""^^ . a ; rdfs:label "location parameter of Johnson-SL-1" ; "location parameter of Johnson-SL-1"^^ ; "JohnsonSL1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location parameter"^^ ; "location"^^ . a ; rdfs:label "median of Categorical Nonordered 1" ; "median of Categorical Nonordered 1"^^ ; "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"^^ . a ; rdfs:label "CDF of Log-Normal 2" ; "CDF of Log-Normal 2"^^ ; "\\frac12 + \\frac12\\,\\text{erf}\\Big[\\frac{\\log x-\\mu}{\\sqrt{2}\\sqrt{var}}\\Big]"^^ ; "1/2 + 1/2 * erf( (log(x)-mu) / (sqrt(2)*sqrt(var)) )"^^ . a ; rdfs:label "SF of Standard Uniform 1" ; "SF of Standard Uniform 1"^^ ; "1-x"^^ ; "1-x"^^ . a owl:DatatypeProperty ; rdfs:label "has MathML expression" . a ; rdfs:label "relationship between Cauchy 1 and Standard Cauchy 1 whereby x0=0, \\\\gamma=1" ; ; ; "x0=0, \\gamma=1"^^ ; "Cauchy1(x_0,\\gamma) \\rightarrow StandardCauchy1"^^ ; "\\url{https://en.wikipedia.org/wiki/Cauchy_distribution}"^^ . a ; rdfs:label "shape of Log-Normal-6" ; "shape of Log-Normal-6"^^ ; "LogNormal6.geomStdev"^^ ; "\\sigma_g"^^ ; "\\sigma_g > 0"^^ ; "shape"^^ ; "geomStdev"^^ . a ; rdfs:label "shape of Pareto-Type-I" ; "shape of Pareto-Type-I"^^ ; "ParetoTypeI1.shape"^^ ; "\\alpha"^^ ; "\\alpha > 0, \\alpha \\in R"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "mean of Generalized Poisson 1" ; "mean of Generalized Poisson 1"^^ ; "\\frac{\\theta}{1 -\\delta}"^^ . a ; rdfs:label "CDF of Log-Normal 4" ; "CDF of Log-Normal 4"^^ ; "\\frac12 + \\frac12\\,\\text{erf}\\Big[\\frac{\\log x-\\log m}{\\sqrt{2}\\sqrt{\\log(cv^2+1)}}\\Big]"^^ ; "1/2 + 1/2 * erf( (log(x)-log(m)) / (sqrt(2*log(cv^2+1))) )"^^ . a ; rdfs:label "mean of Wishart 1" ; "mean of Wishart 1"^^ ; "n V"^^ . a ; rdfs:label "variance of Categorical Ordered 1" ; "variance of Categorical Ordered 1"^^ ; "Var([x=i]) = p_i (1-p_i); \\quad Cov([x=i],[x=j]) = - p_i p_j~~(i\\neq j)"^^ . a ; rdfs:label "mean of Zero-Inflated-Negative-Binomial-1" ; "mean of Zero-Inflated-Negative-Binomial-1"^^ ; "ZeroInflatedNegativeBinomial1.rate"^^ ; "\\lambda"^^ ; "\\lambda > 0"^^ ; "mean"^^ ; "rate"^^ . a ; rdfs:label "relationship between Bernoulli 1 and Binomial 1 whereby \\\\Sigma X iid" ; ; ; "\\Sigma X (iid)"^^ ; "Bernoulli1(p) \\rightarrow Binomial1(n,p)"^^ ; """\\cite{Leemis:2008tg} \\\\ \\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/BernoulliBinomial.pdf}"""^^ . a owl:Class ; rdfs:label "random variable" ; rdfs:subClassOf . a ; rdfs:label "Johnson SL 1" ; "Johnson SL 1"^^ ; "JohnsonSL1"^^ ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , , , . a ; rdfs:label "mode of Normal 2" ; "mode of Normal 2"^^ ; "\\mu"^^ . a ; rdfs:label "variance of Standard Triangular 1" ; "variance of Standard Triangular 1"^^ ; "42375"^^ . a ; rdfs:label "CDF of Weibull 2" ; "CDF of Weibull 2"^^ ; "1- \\exp(-x^v \\lambda)"^^ ; "1- exp(-x^v * lambda)"^^ . a ; rdfs:label "CDF of GeneralizedPoisson2" ; "CDF of GeneralizedPoisson2"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "Gompertz 1" ; "Gompertz 1"^^ ; "Gompertz1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , . a ; rdfs:label "shape2 of Generalized-Negative-Binomial-1" ; "shape2 of Generalized-Negative-Binomial-1"^^ ; "GeneralizedNegativeBinomial1.m"^^ ; "m"^^ ; "m > 0"^^ ; "shape2"^^ ; "m"^^ . a ; rdfs:label "shape parameter of Muth-1" ; "shape parameter of Muth-1"^^ ; "Muth1.shape"^^ ; "\\kappa"^^ ; "0 \\leq \\kappa \\leq 1"^^ ; "shape parameter"^^ ; "shape"^^ . a ; rdfs:label "PDF of Frechet 2" ; "PDF of Frechet 2"^^ ; "\\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))"^^ . a ; rdfs:label "variance of Normal 2" ; "variance of Normal 2"^^ ; "v"^^ . a ; rdfs:label "number of values of Uniform-Discrete-2" ; "number of values of Uniform-Discrete-2"^^ ; "UniformDiscrete2.numberOfValues"^^ ; "n"^^ ; "n \\in N"^^ ; "number of values"^^ ; "numberOfValues"^^ . a owl:ObjectProperty ; rdfs:label "distribution has CDF" . a ; rdfs:label "mean of Nakagami 1" ; "mean of Nakagami 1"^^ ; "\\frac{\\Gamma(m+\\frac{1}{2})}{\\Gamma(m)} \\Big(\\frac{\\Omega}{m}\\Big)^{\\frac12}"^^ . a ; rdfs:label "Exponential 1" , "Negative exponential"^^ ; "Exponential 1"^^ ; "Exponential1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in [0,+\\infty)"^^ ; . a ; rdfs:label "median of Maxwell Boltzmann 1" ; "median of Maxwell Boltzmann 1"^^ ; "\\sqrt{2} a"^^ . a ; rdfs:label "variance of Beta 1" ; "variance of Beta 1"^^ ; "\\frac{\\alpha\\beta}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}"^^ . a ; rdfs:label "dispersion of Generalized-Poisson-1" ; "dispersion of Generalized-Poisson-1"^^ ; "GeneralizedPoisson1.dispersion"^^ ; "\\delta"^^ ; "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"^^ ; "dispersion"^^ ; "dispersion"^^ . a ; rdfs:label "scale parameter of Johnson-SU-1" ; "scale parameter of Johnson-SU-1"^^ ; "JohnsonSU1.scale"^^ ; "\\sigma"^^ ; "\\sigma > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label """relationship between Normal 2 and Normal 3 whereby \\\\mu = \\\\mu, \\\\tau = 1 / v""" ; ; ; """\\mu = \\mu, \\tau = 1 / v"""^^ ; "Normal2(\\mu,v) \\rightarrow Normal3(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "category probabilities of Categorical-Ordered-1" ; "category probabilities of Categorical-Ordered-1"^^ ; "CategoricalOrdered1.categoryProb"^^ ; "p_1, \\ldots, p_k"^^ ; "0 \\leq p_i \\leq 1, \\Sigma p_i = 1"^^ ; "category probabilities"^^ ; "categoryProb"^^ . a ; rdfs:label "PDF of Exponential 1" ; "PDF of Exponential 1"^^ ; "\\lambda e^{-\\lambda x}"^^ ; "lambda*exp(-lambda*x)"^^ . a ; rdfs:label "variance of Zipf 1" ; "variance of Zipf 1"^^ ; "(H_{n,\\alpha-2} H_{n,\\alpha} - H_{n,\\alpha-1}^2)/ H_{n,\\alpha}^2"^^ . a ; rdfs:label "mean of Multinomial 1" ; "mean of Multinomial 1"^^ ; "E\\{X_i\\} = np_i"^^ . a ; rdfs:label """relationship between Normal 1 and Normal 2 whereby \\\\mu = \\\\mu, v = \\\\sigma^2 """ ; ; ; """\\mu = \\mu, v = \\sigma^2 """^^ ; "Normal1(\\mu,\\sigma) \\rightarrow Normal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "variance of Epanechnikov 1" ; "variance of Epanechnikov 1"^^ ; "\\frac15"^^ . a ; rdfs:label "mean of Zero-Inflated Negative Binomial 1" ; "mean of Zero-Inflated Negative Binomial 1"^^ ; "\\mu"^^ . a owl:Class ; rdfs:label "variate" ; rdfs:subClassOf . a ; rdfs:label "mean of Dirichlet 1" ; "mean of Dirichlet 1"^^ ; "E[X_i] = \\frac{\\alpha_i}{\\sum_k \\alpha_k}"^^ . a ; rdfs:label "PDF of Gompertz 1" ; "PDF of Gompertz 1"^^ ; "b\\eta e^{bx}e^{\\eta}\\exp\\left(-\\eta e^{bx} \\right)"^^ ; "b*eta*exp(b*x)*exp(eta)*exp(-eta*exp(b*x))"^^ . a ; rdfs:label "shape of LKJ-Correlation-2" ; "shape of LKJ-Correlation-2"^^ ; "LKJCorrelation2.shape"^^ ; "\\eta"^^ ; "\\eta > 0"^^ ; "shape "^^ ; "shape"^^ . a ; rdfs:label "mode of Standard Triangular 1" ; "mode of Standard Triangular 1"^^ ; "0"^^ . a ; rdfs:label "HF of Weibull 2" ; "HF of Weibull 2"^^ ; "v\\lambda \\,x^{v-1}"^^ ; "v*lambda * x^(v-1)"^^ . a ; rdfs:label "Inverse Weibull"^^ , "Frechet 2" ; "Frechet 2"^^ ; "Frechet2"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in (m, \\infty)"^^ ; , , . a ; rdfs:label "mean of GeneralizedPoisson2" ; "mean of GeneralizedPoisson2"^^ ; "\\mu"^^ . a ; rdfs:label "relationship between Inverse Gaussian 1 and Standard Normal 1 whereby \\\\lambda \\\\rightarrow \\\\infty " ; ; ; "\\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}"""^^ . a ; rdfs:label "Generalized Poisson 3" ; "Generalized Poisson 3"^^ ; "GeneralizedPoisson3"^^ ; ; ; ; ; "y"^^ ; "y \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "median of Nakagami 1" ; "median of Nakagami 1"^^ ; "\\sqrt{\\Omega}"^^ . a owl:ObjectProperty ; rdfs:label "distribution has PMF" . a ; rdfs:label "mean of Maxwell Boltzmann 1" ; "mean of Maxwell Boltzmann 1"^^ ; "2a \\sqrt{\\frac{2}{\\pi}}"^^ . a ; rdfs:label "mode of Beta 1" ; "mode of Beta 1"^^ ; "\\frac{\\alpha-1}{\\alpha+\\beta-2}"^^ . a ; rdfs:label "lambda of Normal-inverse-gamma-1" ; "lambda of Normal-inverse-gamma-1"^^ ; "NormalInverseGamma1.lambda"^^ ; "\\lambda"^^ ; "\\lambda > 0, \\lambda \\in R"^^ ; "lambda"^^ ; "lambda"^^ . a ; rdfs:label "Poisson intensity of Generalized-Poisson-1" ; "Poisson intensity of Generalized-Poisson-1"^^ ; "GeneralizedPoisson1.rate"^^ ; "\\theta"^^ ; "\\theta > 0"^^ ; "Poisson intensity"^^ ; "rate"^^ . a ; rdfs:label "shape1 of Generalized-Negative-Binomial-1" ; "shape1 of Generalized-Negative-Binomial-1"^^ ; "GeneralizedNegativeBinomial1.beta"^^ ; "\\beta"^^ ; "\\beta = 0 \\text{ or } 1 \\le \\beta \\le \\theta^{-1}"^^ ; "shape1"^^ ; "beta"^^ . a ; rdfs:label "location parameter of Johnson-SU-1" ; "location parameter of Johnson-SU-1"^^ ; "JohnsonSU1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location parameter"^^ ; "location"^^ . a ; rdfs:label "PDF of Wishart 1" ; "PDF of Wishart 1"^^ ; "\\frac{|X|^{\\frac{n-p-1}{2}} e^{-\\frac{\\text{tr}(V^{-1}X)}{2}}}{2^\\frac{np}{2}|V|^\\frac{n}{2}\\Gamma_p(\\frac{n}{2})}"^^ . a ; rdfs:label "PDF of Generalized Gamma 3" ; "PDF of Generalized Gamma 3"^^ ; "\\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)"^^ . a ; rdfs:label "variance of Multinomial 1" ; "variance of Multinomial 1"^^ ; """Var(X_i) = n p_i (1-p_i); \\quad Cov(X_i,X_j) = - n p_i p_j~~(i\\neq j)"""^^ . a ; rdfs:label "relationship between Weibull 1 and Exponential 1 whereby k=1, \\\\lambda_{Exponetial} = 1/\\\\lambda" ; ; ; "k=1, \\lambda_{Exponetial} = 1/\\lambda"^^ ; "Weibull1(\\lambda,k) \\rightarrow Exponential1(\\lambda_{Exponetial})"^^ ; "\\cite{forbes2011statistical}"^^ . a ; rdfs:label "Dirichlet 1" , "Dirichlet"^^ ; "Dirichlet 1"^^ ; "Dirichlet1"^^ ; ; ; ; ; "x"^^ ; "x_1, \\cdots, x_K \\quad\\text{where}\\quad x_i \\in [0,1]\\quad and \\quad\\sum_{i=1}^K x_i = 1"^^ ; . a ; rdfs:label "mode of Log-Normal 2" ; "mode of Log-Normal 2"^^ ; "e^{\\mu-v}"^^ . a ; rdfs:label "non-standardized Student's t-distribution"^^ , "t Location-Scale"^^ , "Student's t-distribution 3" ; "Student's t-distribution 3"^^ ; "StudentT3"^^ ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , , . a ; rdfs:label "location of Levy-1" ; "location of Levy-1"^^ ; "Levy1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "Weibull 2" ; "Weibull 2"^^ ; "Weibull2"^^ ; ; ; ; ; "x"^^ ; "x > 0"^^ ; , . a ; rdfs:label "variance of Bernoulli 1" ; "variance of Bernoulli 1"^^ ; "p(1-p)"^^ . a ; rdfs:label "CDF of Johnson SL 1" ; "CDF of Johnson SL 1"^^ ; """\\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)))"""^^ . a ; rdfs:label "CDF of Zero-Inflated Negative Binomial 1" ; "CDF of Zero-Inflated Negative Binomial 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "c(PMF1,cumsum(PMF2)+PMF1)"^^ . a ; rdfs:label "relationship between Standard Uniform 1 and Standard Power 1 whereby \\\\text{If } X \\\\sim \\\\text{ StandardUniform1, then } Y = X^{1/\\\\beta} \\\\text{ has the StandardPower}\\\\beta \\\\text{ distribution, where } \\\\beta > 1" ; ; ; "\\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)"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformStandardpowerB.pdf}"^^ . a ; rdfs:label "Mixture Distribution 1" ; "Mixture Distribution 1"^^ ; "MixtureDistribution1"^^ ; ; . a ; rdfs:label "PMF of Generalized Poisson 3" ; "PMF of Generalized Poisson 3"^^ ; "\\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))"^^ . a ; rdfs:label "PDF of Nakagami 1" ; "PDF of Nakagami 1"^^ ; "\\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)"^^ . a ; rdfs:label "PDF of Standard Power 1" ; "PDF of Standard Power 1"^^ ; "\\beta x^{\\beta-1}"^^ ; "(beta * x^(beta-1))"^^ . a ; rdfs:label "variance of GeneralizedPoisson2" ; "variance of GeneralizedPoisson2"^^ ; "\\frac{\\mu}{(1 -\\delta)^2}"^^ . a ; rdfs:label "mean of Frechet 2" ; "mean of Frechet 2"^^ ; """\\begin{cases} \\ m+\\sigma\\Gamma\\left(1-\\frac{1}{\\alpha}\\right) & \\text{for } \\alpha>1 \\\\ \\ \\infty & \\text{otherwise} \\end{cases}"""^^ . a ; rdfs:label "PDF of Multivariate Gaussian Process Distribution 2" ; "PDF of Multivariate Gaussian Process Distribution 2"^^ ; "\\prod^K_{i=1} \\text{MultivariateNormal1}(x_i|0,w^{-1}_i LL^T)"^^ . a ; rdfs:label "mean of Benford 1" ; "mean of Benford 1"^^ ; "\\approx 3.4402"^^ . a ; rdfs:label "SF of Standard Normal 1" ; "SF of Standard Normal 1"^^ ; "1 - \\frac12 \\text{erfc}\\left(-\\frac{x}{\\sqrt{2}}\\right)"^^ ; "1 - 1/2 *erfc(-x/sqrt(2))"^^ . a ; rdfs:label "Wishart 1" ; "Wishart 1"^^ ; "Wishart1"^^ ; ; ; ; ; "X"^^ ; "X(p \\times p) - \\text{positive definite matrix}"^^ ; , . a ; rdfs:label "relationship between Binomial 1 and Normal 1 whereby \\\\text{For } X \\\\sim Binomial1n,p \\\\text{ as } n \\\\rightarrow \\\\infty, X \\\\text{ is approximately normally distributed } \\\\\\\\Normal1\\\\mu,\\\\sigma \\\\text{ with } \\\\mu=np, \\\\sigma=np1-p." ; ; ; "\\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}"^^ . a ; rdfs:label "variance of Folded Normal 1" ; "variance of Folded Normal 1"^^ ; "\\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"^^ . a ; rdfs:label "degree of freedom of F-1" ; "degree of freedom of F-1"^^ ; "F1.denominator"^^ ; "n_2"^^ ; "n_2 > 0"^^ ; "degree of freedom"^^ ; "denominator"^^ . a ; rdfs:label "CDF of Maxwell Boltzmann 1" ; "CDF of Maxwell Boltzmann 1"^^ ; "\\text{erf}\\left(\\frac{x}{\\sqrt{2} a}\\right) -\\sqrt{\\frac{2}{\\pi}} \\frac{x e^{-x^2/\\left(2a^2\\right)}}{a}"^^ ; "erf(x/(sqrt(2)*a)) -sqrt(2/pi) * (x * exp(-x^2/(2*a^2)) )/a"^^ . a ; rdfs:label "median of Beta 1" ; "median of Beta 1"^^ ; "I_{\\frac{1}{2}}^{[-1]}(\\alpha,\\beta)"^^ . a ; rdfs:label "number of trials of Multinomial-1" ; "number of trials of Multinomial-1"^^ ; "Multinomial1.numberOfTrials"^^ ; "n"^^ ; "n > 0, n \\in N"^^ ; "number of trials"^^ ; "numberOfTrials"^^ . a ; rdfs:label "SF of Beta 1" ; "SF of Beta 1"^^ ; "1-I_x(\\alpha,\\beta)"^^ ; "1 - Rbeta(x, alpha, beta)"^^ . a ; rdfs:label "Standard Two-Sided Power 1" , "STSP"^^ ; "Standard Two-Sided Power 1"^^ ; "StandardTwoSidedPower1"^^ ; ; ; ; ; "x"^^ ; "x \\in (0,1)"^^ ; , . a ; rdfs:label "Generalized Gamma 3" ; "Generalized Gamma 3"^^ ; "GeneralizedGamma3"^^ ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; , , . a ; rdfs:label "scale of Levy-1" ; "scale of Levy-1"^^ ; "Levy1.scale"^^ ; "c"^^ ; "c > 0, c \\in R"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "PDF of Weibull 2" ; "PDF of Weibull 2"^^ ; "v\\lambda \\,x^{v-1} e^{-\\lambda x^{v}}"^^ ; "v*lambda * x^(v-1) * exp(-lambda * x^v)"^^ . a ; rdfs:label "PDF of Dirichlet 1" ; "PDF of Dirichlet 1"^^ ; "\\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)"^^ . a ; rdfs:label "variance of Log-Normal 2" ; "variance of Log-Normal 2"^^ ; "(e^v-1) e^{2\\mu+v}"^^ . a ; rdfs:label "PMF of Zero-Inflated Negative Binomial 1" ; "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"""^^ . a ; rdfs:label "relationship between Negative Binomial 1 and Negative Binomial 5 whereby \\\\alpha = r, \\\\beta = p / 1-p" ; ; ; "\\alpha = r, \\beta = p / (1-p)"^^ ; "NegativeBinomial1(r,p) \\rightarrow NegativeBinomial5(\\alpha, \\beta)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "probability of success of Bernoulli-1" ; "probability of success of Bernoulli-1"^^ ; "Bernoulli1.probability"^^ ; "p"^^ ; "0 ; "probability of success"^^ ; "probability"^^ . a ; rdfs:label "PDF of Johnson SL 1" ; "PDF of Johnson SL 1"^^ ; "\\frac{e^{-\\frac12 \\left( \\gamma + \\delta \\log\\left[\\frac{x-\\mu}{\\sigma}\\right]\\right)^2} \\delta}{\\sqrt{2\\pi} (x-\\mu)}\\\\"^^ ; "exp(-1/2*(gamma+delta*log((x-mu)/sigma))^2)*delta/(sqrt(2*pi)*(x-mu))"^^ . a ; rdfs:label "PDF of Student's t-distribution 3" ; "PDF of Student's t-distribution 3"^^ ; "\\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}}"^^ ; "gamma((nu+1)/2)/gamma(nu/2)*1/sqrt(nu*pi)/sigma*(1+1/nu*((x-mu)/sigma)^2)^(-(nu+1)/2)"^^ . a ; rdfs:label "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}"^^ ; "StandardUniform1 \\rightarrow Benford1"^^ ; "\\url{http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformBenford.pdf}"^^ . a ; rdfs:label "PDF of Maxwell Boltzmann 1" ; "PDF of Maxwell Boltzmann 1"^^ ; "\\sqrt{\\frac{2}{\\pi}} \\frac{x^2 e^{-x^2/\\left(2a^2\\right)}}{a^3}"^^ ; "sqrt(2/pi) * (x^2 * exp(-x^2/(2*a^2)) )/a^3"^^ . a ; rdfs:label "HF of Standard Normal 1" ; "HF of Standard Normal 1"^^ ; "\\frac{\\sqrt{\\frac2\\pi}}{e^{\\frac{x^2}{2}} \\left(2-\\text{erfc}\\left(\\frac{-x}{\\sqrt{2}}\\right)\\right)}"^^ ; "sqrt(2/pi)/(exp(x^2/2)*(2-erfc(-x/sqrt(2))))"^^ . a ; rdfs:label "degree of freedom of F-1" ; "degree of freedom of F-1"^^ ; "F1.numerator"^^ ; "n_1"^^ ; "n_1 > 0"^^ ; "degree of freedom"^^ ; "numerator"^^ . a ; rdfs:label "CDF of Frechet 2" ; "CDF of Frechet 2"^^ ; "\\exp\\Big(-\\Big(\\frac{\\sigma}{x-m}\\Big)^\\alpha\\Big)"^^ ; "exp(-(sigma/(x-m))^alpha)"^^ . a ; rdfs:label "Cholesky parameterization"^^ , "Multivariate Gaussian Process Distribution"^^ , "Multivariate Gaussian Process Distribution 2" ; "Multivariate Gaussian Process Distribution 2"^^ ; "MultivariateGaussianProcess2"^^ ; ; "x"^^ ; "x \\in R^{K\\times N}"^^ ; , . a ; rdfs:label "CDF of Nakagami 1" ; "CDF of Nakagami 1"^^ ; "\\frac{\\gamma(m,\\frac{m}{\\Omega}x^2)}{\\Gamma(m)}"^^ ; "Igamma(m,m/Omega*x^2,lower=T)/gamma(m)"^^ . a ; rdfs:label "Standard Power 1" ; "Standard Power 1"^^ ; "StandardPower1"^^ ; ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,+\\infty)"^^ ; . a ; rdfs:label "success probability of Negative-Binomial-1" ; "success probability of Negative-Binomial-1"^^ ; "NegativeBinomial1.probability"^^ ; "p"^^ ; "p \\in (0,1)"^^ ; "success probability"^^ ; "probability"^^ . a ; rdfs:label "relationship between Negative Binomial 1 and Negative Binomial 4 whereby p = 1 -p" ; ; ; "p = 1 -p"^^ ; "NegativeBinomial1(r,p) \\rightarrow NegativeBinomial4(r,p)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "location of Folded-Normal-1" ; "location of Folded-Normal-1"^^ ; "FoldedNormal1.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "location"^^ ; "location"^^ . a ; rdfs:label "mean of Beta 1" ; "mean of Beta 1"^^ ; "\\frac{\\alpha}{\\alpha+\\beta}"^^ . a ; rdfs:label "Multivariate Normal 1" , "Multivariate Gaussian 1"^^ ; "Multivariate Normal 1"^^ ; "MultivariateNormal1"^^ ; ; ; ; ; ; "x"^^ ; "-\\infty < x_i < \\infty, i = 1,...,k"^^ ; , . a ; rdfs:label "PDF of Mixture Distribution 1" ; "PDF of Mixture Distribution 1"^^ ; "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"^^ . a ; rdfs:label "precision matrix of Multivariate-Student-T-2" ; "precision matrix of Multivariate-Student-T-2"^^ ; "MultivariateStudentT2.precisionMatrix"^^ ; "T"^^ ; "\\text{Inverse of the covariance matrix}"^^ ; "precision matrix"^^ ; "precisionMatrix"^^ . a ; rdfs:label "SF of Standard Triangular 1" ; "SF of Standard Triangular 1"^^ ; """\\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}"""^^ ; """SF1=-1/2*x^2-x+1/2 SF2=1/2*x^2-x+1/2"""^^ . a ; rdfs:label "degrees of freedom of Student-s-t-distribution-3" ; "degrees of freedom of Student-s-t-distribution-3"^^ ; "StudentT3.degreesOfFreedom"^^ ; "\\nu"^^ ; "\\nu > 0, \\nu \\in R"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "median of Lomax 1" ; "median of Lomax 1"^^ ; "\\lambda (\\sqrt[\\alpha]{2} - 1)"^^ . a ; rdfs:label "variance of Standard Normal 1" ; "variance of Standard Normal 1"^^ ; "1"^^ . a ; rdfs:label "relationship between Log-Normal 1 and Log-Normal 3 whereby m = \\\\exp\\\\mu, \\\\sigma = \\\\sigma" ; ; ; "m = \\exp(\\mu), \\sigma = \\sigma"^^ ; "LogNormal1(\\mu,\\sigma) \\rightarrow LogNormal3(m,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "median of Bernoulli 1" ; "median of Bernoulli 1"^^ ; """\\begin{cases} 0 & \\text{if } q > p\\\\ 0.5 & \\text{if } q=p\\\\ 1 & \\text{if } q

. a ; rdfs:label "minimum of Uniform-Discrete-1" ; "minimum of Uniform-Discrete-1"^^ ; "UniformDiscrete1.minimum"^^ ; "a"^^ ; "a \\in \\{\\dots,-2,-1,0,1,2,3,\\dots\\}"^^ ; "minimum"^^ ; "minimum"^^ . a owl:Class ; rdfs:label "scalar" ; rdfs:subClassOf . a ; rdfs:label "Zero-Inflated Negative Binomial 1" ; "Zero-Inflated Negative Binomial 1"^^ ; "ZeroInflatedNegativeBinomial1"^^ ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , , . a ; rdfs:label "shape1 of Generalized-Gamma-2" ; "shape1 of Generalized-Gamma-2"^^ ; "GeneralizedGamma2.shape1"^^ ; "c"^^ ; "c > 0"^^ ; "shape1"^^ ; "shape1"^^ . a ; rdfs:label "ordered cutpoints of Ordered-Logistic-1" ; "ordered cutpoints of Ordered-Logistic-1"^^ ; "OrderedLogistic1.cutpoint"^^ ; "c"^^ ; "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"^^ ; "ordered cutpoints"^^ ; "cutpoint"^^ . a ; rdfs:label "Levy 1" ; "Levy 1"^^ ; "Levy1"^^ ; ; ; ; ; ; ; "x"^^ ; "x \\in [\\mu,+\\infty)"^^ ; , . a ; rdfs:label "HF of Beta 1" ; "HF of Beta 1"^^ ; "\\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))"^^ . a ; rdfs:label "scale of Generalized-Gamma-3" ; "scale of Generalized-Gamma-3"^^ ; "GeneralizedGamma3.scale"^^ ; "r"^^ ; "r > 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "degrees of freedom of Wishart-1" ; "degrees of freedom of Wishart-1"^^ ; "Wishart1.degreesOfFreedom"^^ ; "n"^^ ; "n > p-1"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "PDF of Epanechnikov 1" ; "PDF of Epanechnikov 1"^^ ; "\\frac34 (1-x^2)"^^ ; "3/4 * (1-x^2)"^^ . a ; rdfs:label "PMF of Mixture Distribution 1" ; "PMF of Mixture Distribution 1"^^ ; "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"^^ . a ; rdfs:label "relationship between Laplace 2 and Laplace 1 whereby b=1/\\\\tau" ; ; ; "b=1/\\tau"^^ ; "Laplace2(\\mu,\\tau) \\rightarrow Laplace1(\\mu,b)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape of Nakagami-1" ; "shape of Nakagami-1"^^ ; "Nakagami1.shape"^^ ; "m"^^ ; "m > 0"^^ ; "shape"^^ ; "shape"^^ . a ; rdfs:label "mean of Generalized Negative Binomial 1" ; "mean of Generalized Negative Binomial 1"^^ ; "m \\theta (1-\\theta \\beta)^{-1}"^^ . a ; rdfs:label "scale of Folded-Normal-1" ; "scale of Folded-Normal-1"^^ ; "FoldedNormal1.scale"^^ ; "\\sigma"^^ ; "\\sigma> 0"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "PDF of Johnson SN 1" ; "PDF of Johnson SN 1"^^ ; "\\frac{e^{-\\frac12 \\left( \\gamma + \\frac{\\delta (x-\\mu)}{\\sigma}\\right)^2} \\delta}{\\sqrt{2\\pi}\\sigma}"^^ ; "exp(-1/2*(gamma+(delta*(x-mu))/sigma)^2)*delta/(sqrt(2*pi)*sigma)"^^ . a ; rdfs:label "mode of Binomial 1" ; "mode of Binomial 1"^^ ; "\\lfloor (n + 1)p \\rfloor \\; or \\; \\lfloor (n + 1)p \\rfloor - 1"^^ . a ; rdfs:label "relationship between Uniform Discrete 1 and Uniform Discrete 2 whereby a = 0, b = n" ; ; ; "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}"""^^ . a ; rdfs:label "Conway-Maxwell-Poisson 1" ; "Conway-Maxwell-Poisson 1"^^ ; "ConwayMaxwellPoisson1"^^ ; ; ; ; ; ; "x"^^ ; "x \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "mode of Frechet 2" ; "mode of Frechet 2"^^ ; "m+\\sigma\\left(\\frac{\\alpha}{1+\\alpha}\\right)^{1/\\alpha}"^^ . a ; rdfs:label "HF of Standard Power 1" ; "HF of Standard Power 1"^^ ; "\\frac{\\beta x^{\\beta-1}}{1 - x^\\beta}"^^ ; "beta * x^(beta-1) / (1 - x^beta)"^^ . a ; rdfs:label "relationship between Logistic 1 and Standard Logistic 1 whereby \\\\mu = 0, s = 1" ; ; ; "\\mu = 0, s = 1"^^ ; "Logistic1(\\mu,s) \\rightarrow StandardLogistic1"^^ . a ; rdfs:label "mode of Bernoulli 1" ; "mode of Bernoulli 1"^^ ; """\\begin{cases} 0 & \\text{if } q > p\\\\ 0, 1 & \\text{if } q=p\\\\ 1 & \\text{if } q < p \\end{cases}"""^^ . a ; rdfs:label "mean of Student-s-t-distribution-3" ; "mean of Student-s-t-distribution-3"^^ ; "StudentT3.location"^^ ; "\\mu"^^ ; "\\mu \\in R"^^ ; "mean"^^ ; "location"^^ . a ; rdfs:label "standard deviation of Log-Normal-7" ; "standard deviation of Log-Normal-7"^^ ; "LogNormal7.stdev"^^ ; "\\sigma_N"^^ ; "\\sigma_N > 0"^^ ; "standard deviation"^^ ; "stdev"^^ . a ; rdfs:label "mode of Lomax 1" ; "mode of Lomax 1"^^ ; "0"^^ . a ; rdfs:label "mode of Standard Normal 1" ; "mode of Standard Normal 1"^^ ; "0"^^ . a ; rdfs:label "degrees of freedom of Multivariate-Student-T-2" ; "degrees of freedom of Multivariate-Student-T-2"^^ ; "MultivariateStudentT2.degreesOfFreedom"^^ ; "k"^^ ; "k \\ge 2"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "HF of Standard Triangular 1" ; "HF of Standard Triangular 1"^^ ; """\\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}"""^^ ; """HF1=(2*(x+1))/(-x^2-2*x+1) for -1 < x < 0 HF2=2/(1-x) for 0 \\leq x < 1"""^^ . a ; rdfs:label "maximum of Uniform-Discrete-1" ; "maximum of Uniform-Discrete-1"^^ ; "UniformDiscrete1.maximum"^^ ; "b"^^ ; "b \\in \\{\\dots,-2,-1,0,1,2,3,\\dots\\}, b \\ge a"^^ ; "maximum"^^ ; "maximum"^^ . a ; rdfs:label "index parameter of Negative-Binomial-3" ; "index parameter of Negative-Binomial-3"^^ ; "NegativeBinomial3.dispersion"^^ ; "\\phi"^^ ; "\\phi \\in R, \\phi > 0"^^ ; "index parameter"^^ ; "dispersion"^^ . a ; rdfs:label "Binomial distribution with logit parameterisation"^^ , "Binomial 2" ; "Binomial 2"^^ ; "Binomial2"^^ ; ; ; "k"^^ ; "k \\in \\{0,\\dots,n\\}"^^ ; , . a ; rdfs:label "PDF of Levy 1" ; "PDF of Levy 1"^^ ; "\\sqrt{\\frac{c}{2\\pi}} \\frac{e^{-\\frac{c}{2(x-\\mu)}}}{(x-\\mu)^{3/2}}"^^ ; "sqrt(c/2/pi) * exp(-c/2/(x-mu)) / (x-mu)^(3/2)"^^ . a ; rdfs:label "predictor of Ordered-Logistic-1" ; "predictor of Ordered-Logistic-1"^^ ; "OrderedLogistic1.predictor"^^ ; "\\eta"^^ ; "\\eta \\in R"^^ ; "predictor"^^ ; "predictor"^^ . a ; rdfs:label "shape2 of Generalized-Gamma-2" ; "shape2 of Generalized-Gamma-2"^^ ; "GeneralizedGamma2.shape2"^^ ; "k"^^ ; "k > 0"^^ ; "shape2"^^ ; "shape2"^^ . a ; rdfs:label "scale matrix of Wishart-1" ; "scale matrix of Wishart-1"^^ ; "Wishart1.scaleMatrix"^^ ; "V"^^ ; "V > 0, p\\times p - \\text{positive definite matrix}"^^ ; "scale matrix"^^ ; "scaleMatrix"^^ . a ; rdfs:label "spread of Nakagami-1" ; "spread of Nakagami-1"^^ ; "Nakagami1.spread"^^ ; "\\Omega"^^ ; "\\Omega > 0"^^ ; "spread"^^ ; "spread"^^ . a ; rdfs:label """relationship between Log-Normal 6 and Log-Normal 5 whereby \\\\mu=\\\\logm, \\\\tau= 1/\\\\log^2 \\\\sigma_g""" ; ; ; """\\mu=\\log(m), \\tau= 1/\\log^2 (\\sigma_g)"""^^ ; "LogNormal6(m,\\sigma_g) \\rightarrow LogNormal5(\\mu,\\tau)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "CDF of Generalized Negative Binomial 1" ; "CDF of Generalized Negative Binomial 1"^^ ; "\\Sigma_{i=1}^x f(i), x \\in \\{0,1,2,...\\} \\text{ with } f \\text{ the PMF}"^^ ; "cumsum(PMF)"^^ . a ; rdfs:label "relationship between Logistic 2 and Standard Logistic 1 whereby \\\\mu = 0, \\\\tau = 1" ; ; ; "\\mu = 0, \\tau = 1"^^ ; "Logistic2(\\mu,\\tau) \\rightarrow StandardLogistic1"^^ . a owl:Class ; rdfs:label "interval" ; rdfs:subClassOf . a ; rdfs:label "median of Binomial 1" ; "median of Binomial 1"^^ ; "\\lfloor np \\rfloor \\text{ or } \\lceil np \\rceil"^^ . a ; rdfs:label "Johnson SN 1" ; "Johnson SN 1"^^ ; "JohnsonSN1"^^ ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , , , . a ; rdfs:label "CDF of Standard Power 1" ; "CDF of Standard Power 1"^^ ; "x^\\beta"^^ ; "x^beta"^^ . a ; rdfs:label "median of Frechet 2" ; "median of Frechet 2"^^ ; "m+\\frac{\\sigma}{\\sqrt[\\alpha]{\\log_e(2)}}"^^ . a ; rdfs:label """relationship between Normal 3 and Normal 2 whereby \\\\mu = \\\\mu, v = 1 / \\\\tau""" ; ; ; """\\mu = \\mu, v = 1 / \\tau"""^^ ; "Normal3(\\mu,\\tau) \\rightarrow Normal2(\\mu,v)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PMF of Conway-Maxwell-Poisson 1" ; "PMF of Conway-Maxwell-Poisson 1"^^ ; "\\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 }}"""^^ . a ; rdfs:label "GeneralizedPoisson2" ; "GeneralizedPoisson2"^^ ; "GeneralizedPoisson2"^^ ; ; ; ; ; "k"^^ ; "k \\in \\{0,1,2,3,\\dots\\}"^^ ; , . a ; rdfs:label "median of Standard Triangular 1" ; "median of Standard Triangular 1"^^ ; "0"^^ . a ; rdfs:label "probability of zero of Zero-Inflated-Negative-Binomial-1" ; "probability of zero of Zero-Inflated-Negative-Binomial-1"^^ ; "ZeroInflatedNegativeBinomial1.probabilityOfZero"^^ ; "p0"^^ ; "0 ; "probability of zero"^^ ; "probabilityOfZero"^^ . a owl:Class ; rdfs:label "mathematical function" ; rdfs:subClassOf . a ; rdfs:label "mean of Negative-Binomial-3" ; "mean of Negative-Binomial-3"^^ ; "NegativeBinomial3.mean"^^ ; "\\mu"^^ ; "\\mu \\in R, \\mu > 0"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label "relationship between Negative Binomial 2 and Negative Binomial 3 whereby \\\\mu=\\\\lambda, \\\\phi=1/\\\\tau" ; ; ; "\\mu=\\lambda, \\phi=1/\\tau"^^ ; "NegativeBinomial2(\\lambda, \\tau) \\rightarrow NegativeBinomial3(\\mu, \\phi)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "shape of Generalized-Gamma-3" ; "shape of Generalized-Gamma-3"^^ ; "GeneralizedGamma3.shape2"^^ ; "\\beta"^^ ; "\\beta > 0"^^ ; "shape"^^ ; "shape2"^^ . a ; rdfs:label "mean of x of Log-Normal-7" ; "mean of x of Log-Normal-7"^^ ; "LogNormal7.mean"^^ ; "\\mu_N"^^ ; "\\mu_N \\in R"^^ ; "mean of x"^^ ; "mean"^^ . a ; rdfs:label "scale of Student-s-t-distribution-3" ; "scale of Student-s-t-distribution-3"^^ ; "StudentT3.scale"^^ ; "\\sigma"^^ ; "\\mu \\in R^{+}"^^ ; "scale"^^ ; "scale"^^ . a ; rdfs:label "mean of Epanechnikov 1" ; "mean of Epanechnikov 1"^^ ; "0"^^ . a ; rdfs:label "PMF of Bernoulli 1" ; "PMF of Bernoulli 1"^^ ; """\\begin{cases} q=(1-p) & \\text{for }k=0 \\\\ p & \\text{for }k=1 \\end{cases}"""^^ ; """q=(1-p) for k=0 p for k=1"""^^ . a ; rdfs:label "Multinomial 1" ; "Multinomial 1"^^ ; "Multinomial1"^^ ; ; ; ; "X"^^ ; "X_i \\in \\{0,\\dots,n\\}, \\Sigma X_i = n"^^ ; , . a ; rdfs:label "PMF of Ordered Logistic 1" ; "PMF of Ordered Logistic 1"^^ ; """\\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 . a ; rdfs:label "degrees of freedom of Chi-1" ; "degrees of freedom of Chi-1"^^ ; "Chi1.degreesOfFreedom"^^ ; "k"^^ ; "k > 0"^^ ; "degrees of freedom"^^ ; "degreesOfFreedom"^^ . a ; rdfs:label "CDF of Levy 1" ; "CDF of Levy 1"^^ ; "\\textrm{erfc}\\left(\\sqrt{\\frac{c}{2(x-\\mu)}}\\right)"^^ ; "erfc(sqrt(c/2/(x-mu)))"^^ . a ; rdfs:label "variance of Wishart 1" ; "variance of Wishart 1"^^ ; "Var(X_{ij}) = n \\left (v_{ij}^2+v_{ii}v_{jj} \\right )"^^ . a ; rdfs:label "scale of Generalized-Negative-Binomial-1" ; "scale of Generalized-Negative-Binomial-1"^^ ; "GeneralizedNegativeBinomial1.theta"^^ ; "\\theta"^^ ; "0 < \\theta < 1"^^ ; "scale"^^ ; "theta"^^ . a ; rdfs:label "scale parameter of Maxwell-Boltzmann-1" ; "scale parameter of Maxwell-Boltzmann-1"^^ ; "MaxwellBoltzmann1.scale"^^ ; "a"^^ ; "a > 0"^^ ; "scale parameter"^^ ; "scale"^^ . a ; rdfs:label "relationship between Noncentral F 1 and Noncentral chi-squared 1 whereby F \\\\sim NoncentralF1\\\\lambda,m,n \\\\text{, as } n \\\\rightarrow \\\\infty \\\\text{ then } nF \\\\text{ approaches a non-central chi-square distribution with } m \\\\text{ degrees of freedom and non-central parameter } \\\\lambda" ; ; ; "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)"^^ ; "\\cite{walck2007handbook}"^^ . a ; rdfs:label "shape of Normal-inverse-gamma-1" ; "shape of Normal-inverse-gamma-1"^^ ; "NormalInverseGamma1.alpha"^^ ; "\\alpha"^^ ; "\\alpha > 0, \\alpha \\in R"^^ ; "shape"^^ ; "alpha"^^ . a ; rdfs:label "mixing coefficients of Mixture-Distribution-1" ; "mixing coefficients of Mixture-Distribution-1"^^ ; "MixtureDistribution1.weight"^^ ; "\\pi_1, \\ldots, \\pi_k"^^ ; "\\Sigma_{i=1}^K \\pi_i=1; 0\\le \\pi_i \\le 1"^^ ; "mixing coefficients"^^ ; "weight"^^ . a ; rdfs:label "relationship between Log-Normal 2 and Log-Normal 4 whereby m = \\\\exp\\\\mu, cv = \\\\sqrt{\\\\expv - 1}" ; ; ; "m = \\exp(\\mu), cv = \\sqrt{\\exp(v) - 1}"^^ ; "LogNormal2(\\mu,v) \\rightarrow LogNormal4(m,cv)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "mean of Binomial 1" ; "mean of Binomial 1"^^ ; "np"^^ . a ; rdfs:label "Arcsine 1" ; "Arcsine 1"^^ ; "Arcsine1"^^ ; ; ; ; ; ; ; ; ; "x"^^ ; "x \\in (0,1)"^^ . a ; rdfs:label "mode of Nakagami 1" ; "mode of Nakagami 1"^^ ; "\\frac{\\sqrt{2}}{2} \\Big(\\frac{(2m-1)\\Omega}{m} \\Big)^{1/2}"^^ . a ; rdfs:label "mean of Standard Power 1" ; "mean of Standard Power 1"^^ ; "\\beta/(\\beta+1)"^^ . a ; rdfs:label "relationship between Generalized Negative Binomial 1 and Binomial 1 whereby \\\\beta=0 \\\\text{ and set } m=n, \\\\theta=p" ; ; ; "\\beta=0 \\text{ and set } m=n, \\theta=p"^^ ; "GeneralizedNegativeBinomial1(\\theta,\\beta,m) \\rightarrow Binomial1(n,p)"^^ ; "\\cite{Consul:2006qf}"^^ . a ; rdfs:label "mean of Standard-Normal-1" ; "mean of Standard-Normal-1"^^ ; "StandardNormal1.mean"^^ ; "\\mu"^^ ; "\\mu=0"^^ ; "mean"^^ ; "mean"^^ . a ; rdfs:label """relationship between Log-Normal 6 and Log-Normal 1 whereby \\\\mu=\\\\logm, \\\\sigma=\\\\log\\\\sigma_g""" ; ; ; """\\mu=\\log(m), \\sigma=\\log(\\sigma_g)"""^^ ; "LogNormal6(m,\\sigma_g) \\rightarrow LogNormal1(\\mu,\\sigma)"^^ ; "ProbOnto spec"^^ . a ; rdfs:label "PMF of GeneralizedPoisson2" ; "PMF of GeneralizedPoisson2"^^ ; "\\frac{\\mu (1-\\delta) [\\mu(1-\\delta)+\\delta k]^{k-1}}{k!} e^{-[\\mu (1-\\delta) + \\delta k]}"^^ ; "-(mu*(1-delta)*(mu*(1-delta)+delta*k)^(k-1)) / factorial(k) * exp(-(mu*(1-delta)+delta*k))"^^ . a ; rdfs:label "mean of Standard Triangular 1" ; "mean of Standard Triangular 1"^^ ; "0"^^ . a ; rdfs:label "overdispersion of Zero-Inflated-Negative-Binomial-1" ; "overdispersion of Zero-Inflated-Negative-Binomial-1"^^ ; "ZeroInflatedNegativeBinomial1.overdispersion"^^ ; "\\tau"^^ ; "overdispersion"^^ ; "overdispersion"^^ . a owl:Class ; rdfs:label "matrix" ; rdfs:subClassOf . a ; rdfs:label "variance of Log-Normal 7" ; "variance of Log-Normal 7"^^ ; "\\sigma_N^2"^^ . a ; rdfs:label "shape of Generalized-Gamma-3" ; "shape of Generalized-Gamma-3"^^ ; "GeneralizedGamma3.shape1"^^ ; "\\mu"^^ ; "\\mu > 0"^^ ; "shape"^^ ; "shape1"^^ . a ; rdfs:label "CDF of Epanechnikov 1" ; "CDF of Epanechnikov 1"^^ ; "-\\frac14 x^3 + \\frac34 x + \\frac12"^^ ; "-1/4*x^3 + 3/4*x + 1/2"^^ . a ; rdfs:label "mean of Bernoulli 1" ; "mean of Bernoulli 1"^^ ; "p"^^ . a ; rdfs:label "PMF of Multinomial 1" ; "PMF of Multinomial 1"^^ ; "\\frac{n!}{x_1!\\cdots x_k!} p_1^{x_1} \\cdots p_k^{x_k}"^^ . a ; rdfs:label "variance of Chi 1" ; "variance of Chi 1"^^ ; "k-\\mu^2"^^ . a ; rdfs:label "mode of Wishart 1" ; "mode of Wishart 1"^^ ; "(n-p-1) V \\text{ for } n\\leq p+1"^^ . a ; rdfs:label "mean of Levy 1" ; "mean of Levy 1"^^ ; "\\infty"^^ . a ; rdfs:label "t-distribution"^^ , "Student's t-distribution 2" ; "Student's t-distribution 2"^^ ; "StudentT2"^^ ; ; ; ; ; "x"^^ ; "x \\in (-\\infty,+\\infty)"^^ ; , , . a ; rdfs:label "variance of Generalized Negative Binomial 1" ; "variance of Generalized Negative Binomial 1"^^ ; "m \\theta (1-\\theta) (1-\\theta \\beta)^{-3}"^^ . a ; rdfs:label "scale of Normal-inverse-gamma-1" ; "scale of Normal-inverse-gamma-1"^^ ; "NormalInverseGamma1.beta"^^ ; "\\beta"^^ ; "\\beta > 0, \\beta \\in R"^^ ; "scale"^^ ; "beta"^^ . a ; rdfs:label "relationship between Noncentral Beta 1 and Beta 1 whereby \\\\lambda = 0" ; ; ; "\\lambda = 0"^^ ; "NoncentralBeta1(\\lambda, \\alpha,\\beta) \\rightarrow Beta1(\\alpha,\\beta)"^^ ; "\\cite{walck2007handbook}"^^ . a ; rdfs:label "variance of Maxwell Boltzmann 1" ; "variance of Maxwell Boltzmann 1"^^ ; "\\frac{a^2(3 \\pi - 8)}{\\pi}"^^ . a ; rdfs:label "CDF of Johnson SN 1" ; "CDF of Johnson SN 1"^^ ; "\\frac12 \\text{erfc}\\left[ -\\frac{\\delta + \\frac{\\delta (x-\\mu)}{\\sigma}}{\\sqrt{2}}\\right]"^^ ; "1/2 *erfc(-(gamma+ (delta*(x-mu)/sigma))/sqrt(2))"^^ . a ; rdfs:label "SF of Standard Power 1" ; "SF of Standard Power 1"^^ ; "1- x^\\beta"^^ ; "1- x^beta"^^ . a ; rdfs:label "PDF of Arcsine 1" ; "PDF of Arcsine 1"^^ ; "\\frac{1}{\\pi\\sqrt{x(1-x)}}"^^ ; "1/pi/sqrt(x*(1-x))"^^ . a ; rdfs:label "variance of Nakagami 1" ; "variance of Nakagami 1"^^ ; "\\Omega \\Big(1-\\frac{1}{m}\\Big(\\frac{\\Gamma(m+\\frac{1}{2})}{\\Gamma(m)}\\Big)^2 \\Big)"^^ . a ; rdfs:label "variance of Frechet 2" ; "variance of Frechet 2"^^ ; """\\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}"""^^ . a ; rdfs:label "relationship between Negative Binomial 4 and Negative Binomial 1 whereby p = 1 -p" ; ; ; "p = 1 -p"^^ ; "NegativeBinomial4(r,p) \\rightarrow NegativeBinomial1(r,p)"^^ ; "ProbOnto spec"^^ .