% % plain-TeX file % \ifx\pdfoutput\undefined \input mbboard \input mathabx \fi \input typofrmt \input typotabl \useoptions{magstep1,a4,english,preprint} \title{Probability Distributions} \author{Anthony Phan, \today} \maketitle \overfullrule=0pt \def\description{\medbreak\bgroup \def\item##1{\medbreak\hangindent\parindent\leavevmode \hskip-\parindent{{\tt##1}.}\enspace\ignorespaces}% \def\subitem##1{\smallbreak\hangindent2\parindent\leavevmode {\it##1.}\enspace\ignorespaces}% \def\subsubitem##1{\par\hangindent3\parindent\leavevmode \hskip\parindent{##1.}\enspace\ignorespaces}% \let\itemitem=\subitem}% \def\enddescription{\egroup\medbreak}% \def\cs#1{\hbox{\tt\string#1}}% \def\var#1{\ifmmode#1\else$#1$\fi}% usual variable \def\vari#1{\ifmmode\hbox{\it#1}\else{\it#1}\fi}% variable in italic \def\vartype#1{\ifmmode\hbox{\tt#1}\else{\tt#1}\fi}% variable type \section{Continuous Probability distributions\footnote{{\it See}\/ \cs{probability-distributions.h}}} \description \item{lnGamma(\vartype{double} $x$)} Logarithm of the Eulerian Gamma function, $\ln(\Gamma(x))$. \item{Gamma(\vartype{double} $x$)} Eulerian Gamma function $$ \mathbb R_+^*\longrightarrow[\sqrt \pi,+\infty\mathclose[,\qquad x\longmapsto \Gamma(x) =\int_0^{+\infty}t^{x-1}\,{\rm e}^{-t}\,{\rm d}t. $$ Remember that for $n\in\mathbb N$, $n!=\Gamma(n+1)$. \item{Beta(\vartype{double} $x$, \vartype{double} $y$)} Eulerian Beta function $$ (\mathbb R_+^*)^2\longrightarrow\mathbb R_+^*,\qquad x\longmapsto {\rm B}(x, y)=\int_{0}^{1}u^{x-1}(1-u)^{y-1}\,{\rm d}u ={\Gamma(x)\times\Gamma(y)\over\Gamma(x+y)}\,. $$ % Gamma distributions \item{gammapdf(\vartype{double} $x$, \vartype{double} $a$, \vartype{double} $\lambda$)} Probability density function of the Gamma distribution with parameter $a>0$, $\lambda>0$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto \mathbb 1_{\mathbb R_+}(x){\lambda^a\over\Gamma(a)}\, x^{a-1}\,{\rm e}^{-\lambda x}. $$ \item{gammacdf(\vartype{double} $x$, \vartype{double} $a$, \vartype{double} $\lambda$)} Cumulative distribution function of the Gamma distribution with parameter $a>0$, $\lambda>0$. $$ \mathbb R\longrightarrow[0,1\mathclose[,\qquad x\longmapsto\mathbb 1_{\mathbb R_+}(x){\lambda^a\over\Gamma(a)} \int_0^x t^{a-1}\,{\rm e}^{-\lambda t}\,{\rm d}t. $$ \item{gammaicdf(\vartype{double} $p$, \vartype{double} $a$, \vartype{double} $\lambda$)} Inverse cumulative distribution function of the Gamma distribution with parameter $a>0$, $\lambda>0$. $$ [0,1\mathclose[\longrightarrow\mathbb R_+,\qquad p\longmapsto\mathop{\rm gammaicdf}(p,a,\lambda). $$ It is set to $0$ for $p<\cs{accuracy}$ and to \cs{infinity} for $p>1-\cs{accuracy}$. % N(0,1) distribution \item{normlimit} Numerical parameter for normal computations: if $X$ is a random variable with law $\mathcal N(0,1)$, the normal distribution with mean $0$ and standard deviation $1$, then $\mathbb P\{X\geq\cs{normlimit}\}=\mathbb P\{X\leq-\cs{normlimit}\}\approx0$. Its value is (unreasonably) set to $\cs{normlimit} = 10$ ($\cs{normlimit} = 4$ should be sufficient). \item{normalpdf(\vartype{double} $x$)} Probability density function of $\mathcal N(0,1)$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto{\rm e}^{-x^2\!/2}/\sqrt{2\pi}. $$ \item{normalcdf (\vartype{double} $x$)} Cumulative distribution function of $\mathcal N(0,1)$. $$ \mathbb R\longrightarrow\mathopen]0,\mathclose1[,\qquad x\longmapsto\Phi(x)=\int_{-\infty}^x{\rm e}^{-z^2\!/2}\, {{\rm d} z\over\sqrt{2\pi}} ={1\over2}+{1\over\sqrt{2\pi}}\sum_{n=0}^\infty {(-1)^n\,x^{2n+1}\over (2n+1)\times n!}\,. $$ It is computed with its associated power series when $|x|<\cs{normlimit}$, and set to $0$ or $1$ otherwise. \item{normalcdf\_(\vartype{double} $x$)} Cumulative distribution function of $\mathcal N(0,1)$. It is just another implementation of the previous function with a Gamma cumulative distribution function since $$ \Phi(x)= {1\over2}\,\bigl(\mathop{\rm sgn}(x) \times\mathop{\rm gammacdf}(x^2\!/2,1/2)+1,1\bigr), \qquad\hbox{for all $x\in\mathbb R$.} $$ \item{normalicdf(\vartype{double} $p$)} Inverse cumulative distribution function of $\mathcal N(0,1)$. $$ \mathopen]0,1\mathclose[\longrightarrow\mathbb R,\qquad p\longmapsto\mathop{\rm normalicdf}(p)=\Phi^{-1}(p). $$ It is set to $\pm\cs{infinity}$ for $p$ outside of $\mathopen]\cs{accuracy},1-\cs{accuracy}\mathclose[$. Of course, there is also \cs{normalicdf\_}\dots \remark Normal distributions with mean $m$ and standard deviation $\sigma$ are not inplemented since they an be easily derived from the standard normal distribution. For instance, one can set \verbatim double gaussianpdf(double x, double m, double sigma){ return normalpdf((x-m)/sigma)/sigma;} double gaussiancdf(double x, double m, double sigma){ return normalcdf((x-m)/sigma);} double gaussianicdf(double p, double m, double sigma){ return sigma*normalicdf(p)+m;} \endverbatim in order to get the probability, cumulative, inverse cumulative distribution functions of the $\mathcal N(m,\sigma^2)$ distribution with $m\in\mathbb R$ and $\sigma>0$. \endremark % $\chi^2$ distributions \item{chisquarepdf(\vartype{double} $x$, \vartype{double} $\nu$)} Probability density function of $\chi^2(\nu)$, the chi-square (Pearson) distribution with $\nu>0$ degrees of freedom. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto\mathbb 1_{\mathbb R_+}(x) \,{x^{\nu/2-1}{\rm e}^{-x/2}\over 2^{\nu/2}\,\Gamma(\nu/2)}\,. $$ \item{chisquarecdf(\vartype{double} $x$, \vartype{double} $\nu$)} Cumulative distribution function of $\chi^2(\nu)$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto\mathbb 1_{\mathbb R_+}(x) \int_0^x\,{t^{\nu/2-1}{\rm e}^{-t/2} \,{{\rm d}t\over 2^{\nu/2}\,\Gamma(\nu/2)}}\,. $$ \item{chisquareicdf(\vartype{double} $p$, \vartype{double} $\nu$)} Inverse cumulative distribution function of $\chi^2(\nu)$. $$ [0,1\mathclose[\longrightarrow\mathbb R_+,\qquad p\longmapsto\mathop{\rm chisquareicdf}(p,\nu). $$ It is set to $0$ for $p<\cs{accuracy}$ and to \cs{infinity} for $p>1-\cs{accuracy}$. % Beta distributions \item{betapdf(\vartype{double} $x$, \vartype{double} $a$, \vartype{double} $b$)} Probability density function of the Beta distribution with parameters $a>à$ and $b>0$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto\mathbb 1_{[0,1]}(x)\,x^{a-1}(1-x)^{b-1}\!/{\rm B}(a,b). $$ \item{betacdf(\vartype{double} $x$, \vartype{double} $a$, \vartype{double} $b$)} Cumulative distribution function of the Beta distribution with parameters $a>0$ and $b>0$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto \int_0^x\mathbb 1_{[0,1]}(u)\,u^{a-1}(1-u)^{b-1} \,{{\rm d}u\over {\rm B}(a,b)}\,. $$ \item{betaicdf(\vartype{double} $p$, \vartype{double} $a$, \vartype{double} $b$)} Inverse cumulative distribution function of the Beta distribution with parameters $a>0$ and $b>0$. $$ [0,1]\longrightarrow\mathbb R_+,\qquad p\longmapsto\mathop{\rm betaicdf}(p,a,b). $$ It is set to $0$ for $p<\cs{accuracy}$ and to $1$ for $p>1-\cs{accuracy}$. % Student's (T) distributions \item{studentpdf(\vartype{double} $x$, \vartype{double} $\nu$)} Probability density function of $\mathcal T(\nu)$, the Student distribution with $\nu>0$ degrees of freedom. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto {1\over\sqrt{\nu}\,{\rm B}(\nu/2,1/2)} \biggl(1+{x^2\over \nu}\biggr)^{\!\!-(\nu+1)/2} . $$ \item{studentcdf(\vartype{double} $x$, \vartype{double} $\nu$)} Cumulative distribution function of $\mathcal T(\nu)$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto \int_{-\infty}^x \biggl(1+{z^2\over \nu}\biggr)^{\!\!-(\nu+1)/2} {{\rm d}z\over\sqrt{\nu}\,{\rm B}(\nu/2,1/2)} . $$ \item{studenticdf(\vartype{double} $p$, \vartype{double} $\nu$)} Inverse cumulative distribution function of $\mathcal T(\nu)$. $$ \mathopen]0,1\mathclose[\longrightarrow\mathbb R,\qquad p\longmapsto\mathop{\rm studenticdf}(p,\nu). $$ It is set to $\pm\cs{infinity}$ for $p$ outside of $\mathopen]\cs{accuracy},1-\cs{accuracy}\mathclose[$. % Fisher's (F) distributions \item{fisherpdf(\vartype{double} $x$, \vartype{double} $\nu_1$, \vartype{double} $\nu_2$)} Probability density function of $\mathcal F(\nu_1,\nu_2)$, the Fisher distribution with $\nu_1>0$ and $\nu_2>0$ degrees of freedom ($\nu_1$ is the numerator degree of freedom, $\nu_2$ the demominator degree of freedom). $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto \mathbb 1_{\mathbb R_+}(x)\,{(\nu_1/\nu_2)^{\nu_1/2}\,x^{\nu_1/2-1} \over {\rm B}(\nu_1/2,\nu_2/2)(1+x\times \nu_1/\nu_2)^{(\nu_1+\nu_2)/2}}. $$ \item{fishercdf(\vartype{double} $x$, \vartype{double} $\nu_1$, \vartype{double} $\nu_2$)} Cumulative distribution function of $\mathcal F(\nu_1,\nu_2)$. $$ \mathbb R\longrightarrow\mathbb R_+,\qquad x\longmapsto \mathbb 1_{\mathbb R_+}(x)\int_0^x {(\nu_1/\nu_2)^{\nu_1/2}\,z^{\nu_1/2-1} \over {\rm B}(\nu_1/2,\nu_2/2)(1+z\times \nu_1/\nu_2)^{(\nu_1+\nu_2)/2}} \,{\rm d}z. $$ \item{fishericdf(\vartype{double} $p$, \vartype{double} $\nu_1$, \vartype{double} $\nu_2$)} Inverse cumulative distribution function of $\mathcal F(\nu_1, \nu_2)$. $$ [0,1\mathclose[\longrightarrow\mathbb R_+,\qquad p\longmapsto\mathop{\rm fishericdf}(p,\nu_1,\nu_2). $$ It is set to $0$ for $p<\cs{accuracy}$ and to \cs{infinity} for $p>1-\cs{accuracy}$. \section{Discrete Probability distributions\footnote{{\it See}\/ \cs{probability-distributions.h}}} About quantiles, please note that $q_p=k+0.5$ when $F(k)=p$ for $k\in\mathbb N$. % Poisson's distributions \item{poissonpdf(\vartype{double} $x$, \vartype{double} $\lambda$)} Probability distribution function of $\mathcal P(\lambda)$, the Poisson distribution with parameter $\lambda\geq0$. $$ \mathbb R\longrightarrow[0,1],\qquad x\longmapsto\cases{{\rm e}^{-\lambda}\,\lambda^x\!/x!& if $x\in\mathbb N$,\cr 0 &otherwise.} $$ \item{poissoncdf(\vartype{double} $x$, \vartype{double} $\lambda$)} Cumulative distribution function of $\mathcal P(\lambda)$. \item{poissonicdf(\vartype{double} $p$, \vartype{double} $\lambda$)} Inverse cumulative distribution function of $\mathcal P(\lambda)$. % Binomial distributions \item{binomialpdf(\vartype{double} $x$, \vartype{int} $n$, \vartype{double} $\pi$)} Probability distribution function of $\mathcal B(n,\pi)$, the binomial distribution with parameters $n\in\mathbb N^*$ and $\pi\in[0,1]$. $$ \mathbb R\longrightarrow[0,1],\qquad x\longmapsto\cases{C_n^x\,\pi^x(1-\pi)^{n-x}& if $x\in\{0,1,\dots,n\}$,\cr 0 &otherwise.} $$ \item{binomialcdf(\vartype{double} $x$, \vartype{int} $n$, \vartype{double} $\pi$)} Cumulative distribution function of $\mathcal B(n,\pi)$. \item{binomialicdf(\vartype{double} $p$, \vartype{int} $n$, \vartype{double} $\pi$)} Inverse cumulative distribution function of $\mathcal B(n,\pi)$. % Geometric distributions \item{geometricpdf(\vartype{double} $x$, \vartype{double} $\pi$)} Probability distribution function of $\mathcal G(\pi)$, the geometrical distribution with parameter $\pi\in[0,1]$. Describe the law of the first success rank in an infinitely repeated Bernoulli trial with parameter $\pi\in[0,1]$. Thus, it is given by $$ \mathbb R\longrightarrow[0,1], \qquad x\longmapsto \cases{\pi(1-\pi)^{x-1}& if $x\in\{1,2,3,\dots\}$,\cr 0 & otherwise.} $$ \item{geometriccdf(\vartype{double} $x$, \vartype{double} $\pi$)} Cumulative distribution function of $\mathcal G(\pi)$. It returns the sum up to $x$ of the previous probabilities. Thus it is given by $$ \mathbb R\longrightarrow[0,1], \qquad x\longmapsto \cases{1-(1-\pi)^{\mathop{\rm floor}x}& if $x\geq1$,\cr 0 & otherwise.} $$ \item{geometricicdf(\vartype{double} $p$, \vartype{double} $\pi$)} Inverse cumulative distribution function of $\mathcal G(\pi)$. $$ [0,1]\longrightarrow\{1,1.5,2,2.5,3,3.5,\dots\}, \qquad p\longmapsto\mathop{\rm geometricicdf}(p,\pi). $$ \item{negativebinomialpdf(\vartype{double} $x$, \vartype{int} $n$, \vartype{double} $\pi$)} Probability distribution function of the negative binomial ditribution with parameters $n\in\mathbb N^*$ and $\pi\in[0,1]$. Describe the law of the $n$-th success rank, $n\in\mathbb N^*$, in an infinitely repeated Bernoulli trial with parameter $\pi\in[0,1]$. Thus, it is given by $$ \mathbb R\longrightarrow[0,1],\qquad x\longmapsto \cases{C_{x-1}^{n-1}\,\pi^n(1-\pi)^{x-n}& if $x\in\{n,n+1,n+2,\dots\}$,\cr 0 & otherwise.} $$ One can remark that the negative binomial distributions for $n=1$ are just the corresponding geometric ones. It is certainly better when $n=1$ to use functions related to geometric distributions instead of negative binomial distributions related ones. \item{negativebinomialcdf(\vartype{double} $x$, \vartype{int} $n$, \vartype{double} $\pi$)} Cumulative distribution function of the negative binomial ditribution with parameters $n\in\mathbb N^*$ and $\pi\in[0,1]$. One can easily prove that it is given by $$ \mathbb R\longrightarrow[0,1], \qquad x\longmapsto 1-\mathop{\rm binomialcdf}(n-1,\mathop{\rm floor} x,\pi). $$ \item{negativebinomialicdf(\vartype{double} $p$, \vartype{int} $n$, \vartype{double} $\pi$)} Inverse cumulative distribution function of the negative binomial ditribution with parameters $n\in\mathbb N^*$ and $\pi\in[0,1]$. $$ [0,1]\longrightarrow\{n,n+0.5,n+1,n+1.5,\dots\}, \qquad p\longmapsto\mathop{\rm negativebinomialicdf}(p,n,\pi). $$ % Hypergeometric distributions \item{hypergeometricpdf(\vartype{double} $x$, \vartype{int} $N$, \vartype{int} $n$, \vartype{double} $\pi$)} Probability distribution function of $\mathcal H(N, n,\pi)$, the hypergeometrical distribution with parameters $N\geq n\in\mathbb N^*$ and $\pi\in[0,1]$. One should have $N\pi\in\mathbb N$. $$ \eqalign{ \mathbb R&\longrightarrow[0,1],\cr x&\longmapsto\cases{\displaystyle {C_{N\pi}^x\,C_{N(1-\pi)}^{n-x}\over C_N^n}& if $x\in\{\max(0,n-N(1-\pi)),\dots,\min(n,N\pi)\}$,\cr 0 &otherwise.} } $$ \item{hypergeometriccdf(\vartype{double} $x$, \vartype{int} $N$, \vartype{int} $n$, \vartype{double} $\pi$)} Cumulative distribution function of $\mathcal H(N, n,\pi)$. \item{hypergeometricicdf(\vartype{double} $p$, \vartype{int} $N$, \vartype{int} $n$, \vartype{double} $\pi$)} Inverse cumulative distribution function of $\mathcal H(N,n,\pi)$. \section{Rather specific Probability distributions\footnote{{\it See}\/ \cs{probability-distributions.h}}} \item{Kolmogorovcdf(\vartype{double} $x$, \vartype{int} $n$)} Cumulative distribution function of Kolmogorov distributions. These are the famous probability distributions involved in Kolmogorov--Smirnov (two-sided) Goodness of Fit tests with statistic $$ K_n={\|F-F_n\|}_\infty=\sup_{x\in\mathbb R}\bigl|F(x)-F_n(x)\bigr| =\max\nolimits_{i = 1}^n \bigl(F(X_{(i)})-(i-1)/n\bigr) \vee\bigl(i/n-F(X_{(i)})\bigr). $$ Their computation is based on ``Evaluating Kolmogorov's Distribution'' by George Marsa\-glia and Wai Wan Tsang. \item{kolmogorovicdf(\vartype{double} $p$, \vartype{int} $n$)} Inverse cumulative distribution function of Kolmogorov distributions. (Please do not use it since it is based on the bisection method or dichotomy.) \item{klmcdf(\vartype{double} $x$, \vartype{int} $n$)} Cumulative distribution function of the limiting distributions associated to Kolmogorov distribution by Dudley's asymptotic formula (1964): $$ \lim_{n\to\infty}\mathbb P\bigl\{K_n\leq u/\!\sqrt n\bigr\} =1+2\sum_{k=1}^\infty(-1)^k\exp\bigl(-2k^2u^2\bigr), $$ with some numerical adjustments (Stephens M.A., 1970). \item{klmicdf(\vartype{double} $p$, \vartype{int} $n$)} Inverse cumulative distribution function of the previous distributions. \item{kpmcdf(\vartype{double} $x$, \vartype{int} $n$)} Cumulative distribution function of the distribution involved in the one sided Goodness of Fit tests with statistic $$ K^+_n=\sup_{x\in\mathbb R}\bigl(F(x)-F_n(x)\bigr) =\max\nolimits_{1\leq i\leq n} \Bigl({i\over n}-F\bigl(X_{(i)}\bigr)\Bigr) $$ or $$ K^-_n=\sup_{x\in\mathbb R}\bigl(F_n(x)-F(x)\bigr) =\max\nolimits_{1\leq i\leq n} \Bigl(F\bigl(X_{(i)}\bigr)-{i-1\over n}\Bigr) $$ which share the same distribution: for $x\in[0,1]$, $$ \eqalign{ \hbox{\tt kpmcdf}(x,n) =\Proba\bigl\{K^\pm_n\leq x\bigr\} &=x\sum_{0\leq k\leq nx}{n\choose k}(k/n-x)^k(x+1-k/n)^{n-k-1}\cr &=1-x\sum_{nx