Template:Infobox probability distribution/testcases

Normal distribution

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Normal Distribution Probability density function Probability density function for the normal distribution The red curve is the standard normal distribution Cumulative distribution function Cumulative distribution function for the normal distribution Notation ${𝒩}(\mu ,{\sigma }^2)$ Parameters $\mu \in \mathbb{ℝ}$ = mean (location) ${\sigma }^2>0$ = variance (squared scale) Support $x\in \mathbb{ℝ}$ PDF $\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$ CDF $\frac{1}{2}[1+\mathrm{erf}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)]$ Quantile $\mu +\sigma \sqrt{2}{\mathrm{erf}}^{-1}(2p-1)$ Mean $\mu$ Median $\mu$ Mode $\mu$ Variance ${\sigma }^2$ MAD $\sqrt{2/\pi }\sigma$ Skewness $0$ Excess kurtosis $0$ Entropy $\frac{1}{2}\log (2\pi e{\sigma }^2)$ MGF $\exp (\mu t+{\sigma }^2{t}^2/2)$ CF $\exp (i\mu t-{\sigma }^2{t}^2/2)$ Fisher information ${ℐ}(\mu ,\sigma )=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 2/{\sigma }^2\end{array}\right)$ ${ℐ}(\mu ,{\sigma }^2)=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 1/(2{\sigma }^4)\end{array}\right)$ Kullback–Leibler divergence $D_{\text{KL}}({{𝒩}}_0\Vert {{𝒩}}_1)=\frac{1}{2}\{({\sigma }_0/{\sigma }_1{)}^2+\frac{({\mu }_1-{\mu }_0{)}^2}{{\sigma }_1^2}-1+2\ln \frac{{\sigma }_1}{{\sigma }_0}\}$

Normal Distribution Probability density function Probability density function for the normal distribution The red curve is the standard normal distribution Cumulative distribution function Cumulative distribution function for the normal distribution Notation ${𝒩}(\mu ,{\sigma }^2)$ Parameters $\mu \in \mathbb{ℝ}$ = mean (location) ${\sigma }^2>0$ = variance (squared scale) Support $x\in \mathbb{ℝ}$ PDF $\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$ CDF $\frac{1}{2}[1+\mathrm{erf}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)]$ Quantile $\mu +\sigma \sqrt{2}{\mathrm{erf}}^{-1}(2p-1)$ Mean $\mu$ Median $\mu$ Mode $\mu$ Variance ${\sigma }^2$ MAD $\sqrt{2/\pi }\sigma$ Skewness $0$ Ex. kurtosis $0$ Entropy $\frac{1}{2}\log (2\pi e{\sigma }^2)$ MGF $\exp (\mu t+{\sigma }^2{t}^2/2)$ CF $\exp (i\mu t-{\sigma }^2{t}^2/2)$ Fisher information ${ℐ}(\mu ,\sigma )=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 2/{\sigma }^2\end{array}\right)$ ${ℐ}(\mu ,{\sigma }^2)=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 1/(2{\sigma }^4)\end{array}\right)$ Kullback-Leibler divergence $D_{\text{KL}}({{𝒩}}_0\Vert {{𝒩}}_1)=\frac{1}{2}\{({\sigma }_0/{\sigma }_1{)}^2+\frac{({\mu }_1-{\mu }_0{)}^2}{{\sigma }_1^2}-1+2\ln \frac{{\sigma }_1}{{\sigma }_0}\}$

Binomial distribution

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Binomial distribution Probability mass function Probability mass function for the binomial distribution Cumulative distribution function Cumulative distribution function for the binomial distribution Notation $B(n,p)$ Parameters $n\in \{0,1,2,\dots \}$ – number of trials $p\in [0,1]$ – success probability for each trial Support $k\in \{0,1,\dots ,n\}$ – number of successes PMF $(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$ CDF $I_{1-p}(n-k,1+k)$ Mean $np$ Median $\lfloor np\rfloor$ or $\lceil np\rceil$ Mode $\lfloor (n+1)p\rfloor$ or $\lceil (n+1)p\rceil -1$ Variance $np(1-p)$ Skewness $\frac{1-2p}{\sqrt{np(1-p)}}$ Excess kurtosis $\frac{1-6p(1-p)}{np(1-p)}$ Entropy $\frac{1}{2}{\log }_2(2\pi enp(1-p))+O\left(\frac{1}{n}\right)$ in shannons. For nats, use the natural log in the log. MGF $(1-p+p{e}^t{)}^n$ CF $(1-p+p{e}^{it}{)}^n$ PGF $G(z)=[(1-p)+pz{]}^n$ Fisher information $g_n(p)=\frac{n}{p(1-p)}$ (for fixed $n$)

Binomial distribution Probability mass function Probability mass function for the binomial distribution Cumulative distribution function Cumulative distribution function for the binomial distribution Notation $B(n,p)$ Parameters $n\in \{0,1,2,\dots \}$ – number of trials $p\in [0,1]$ – success probability for each trial Support $k\in \{0,1,\dots ,n\}$ – number of successes PMF $(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k}$ CDF $I_{1-p}(n-k,1+k)$ Mean $np$ Median $\lfloor np\rfloor$ or $\lceil np\rceil$ Mode $\lfloor (n+1)p\rfloor$ or $\lceil (n+1)p\rceil -1$ Variance $np(1-p)$ Skewness $\frac{1-2p}{\sqrt{np(1-p)}}$ Ex. kurtosis $\frac{1-6p(1-p)}{np(1-p)}$ Entropy $\frac{1}{2}{\log }_2(2\pi enp(1-p))+O\left(\frac{1}{n}\right)$ in shannons. For nats, use the natural log in the log. MGF $(1-p+p{e}^t{)}^n$ CF $(1-p+p{e}^{it}{)}^n$ PGF $G(z)=[(1-p)+pz{]}^n$ Fisher information $g_n(p)=\frac{n}{p(1-p)}$ (for fixed $n$)

Geometric distribution

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Geometric Probability mass function File:Geometric pmf.svg Cumulative distribution function File:Geometric cdf.svg Parameters $0<p<1$ success probability (real) $0<p\le 1$ success probability (real) Support k trials where $k\in \{1,2,3,\dots \}$ k failures where $k\in \{0,1,2,3,\dots \}$ PMF $(1-p{)}^{k-1}p$ $(1-p{)}^{k}p$ CDF $1-(1-p{)}^k$ $1-(1-p{)}^{k+1}$ Mean $\frac{1}{p}$ $\frac{1-p}{p}$ Median $\lceil \frac{-1}{{\log }_2(1-p)}\rceil$ (not unique if $-1/{\log }_2(1-p)$ is an integer) $\lceil \frac{-1}{{\log }_2(1-p)}\rceil -1$ (not unique if $-1/{\log }_2(1-p)$ is an integer) Mode $1$ $0$ Variance $\frac{1-p}{p^2}$ $\frac{1-p}{p^2}$ Skewness $\frac{2-p}{\sqrt{1-p}}$ $\frac{2-p}{\sqrt{1-p}}$ Excess kurtosis $6+\frac{p^2}{1-p}$ $6+\frac{p^2}{1-p}$ Entropy $\frac{-(1-p){\log }_2(1-p)-p{\log }_{2}p}{p}$ $\frac{-(1-p){\log }_2(1-p)-p{\log }_{2}p}{p}$ MGF $\frac{p{e}^t}{1-(1-p)e^t},$ for $t<-\ln (1-p)$ $\frac{p}{1-(1-p)e^t}$ CF $\frac{p{e}^{it}}{1-(1-p)e^{it}}$ $\frac{p}{1-(1-p)e^{it}}$

Geometric Probability mass function File:Geometric pmf.svg Cumulative distribution function File:Geometric cdf.svg Parameters $0<p<1$ success probability (real) $0<p\le 1$ success probability (real) Support k trials where $k\in \{1,2,3,\dots \}$ k failures where $k\in \{0,1,2,3,\dots \}$ PMF $(1-p{)}^{k-1}p$ $(1-p{)}^{k}p$ CDF $1-(1-p{)}^k$ $1-(1-p{)}^{k+1}$ Mean $\frac{1}{p}$ $\frac{1-p}{p}$ Median $\lceil \frac{-1}{{\log }_2(1-p)}\rceil$ (not unique if $-1/{\log }_2(1-p)$ is an integer) $\lceil \frac{-1}{{\log }_2(1-p)}\rceil -1$ (not unique if $-1/{\log }_2(1-p)$ is an integer) Mode $1$ $0$ Variance $\frac{1-p}{p^2}$ $\frac{1-p}{p^2}$ Skewness $\frac{2-p}{\sqrt{1-p}}$ $\frac{2-p}{\sqrt{1-p}}$ Ex. kurtosis $6+\frac{p^2}{1-p}$ $6+\frac{p^2}{1-p}$ Entropy $\frac{-(1-p){\log }_2(1-p)-p{\log }_{2}p}{p}$ $\frac{-(1-p){\log }_2(1-p)-p{\log }_{2}p}{p}$ MGF $\frac{p{e}^t}{1-(1-p)e^t},$ for $t<-\ln (1-p)$ $\frac{p}{1-(1-p)e^t}$ CF $\frac{p{e}^{it}}{1-(1-p)e^{it}}$ $\frac{p}{1-(1-p)e^{it}}$

Gamma distribution

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Gamma Probability density function Probability density plots of gamma distributions Cumulative distribution function Cumulative distribution plots of gamma distributions Parameters k > 0 shape θ > 0 scale α > 0 shape β > 0 rate Support $x\in (0,\mathrm{\infty })$ $x\in (0,\mathrm{\infty })$ PDF $\frac{1}{\mathrm{\Gamma }(k){\theta }^k}{x}^{k-1}{e}^{-\frac{x}{\theta }}$ $\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{x}^{\alpha -1}{e}^{-\beta x}$ CDF $\frac{1}{\mathrm{\Gamma }(k)}\gamma (k,\frac{x}{\theta })$ $\frac{1}{\mathrm{\Gamma }(\alpha )}\gamma (\alpha ,\beta x)$ Mean $\mathrm{E}[X]=k\theta$ $\mathrm{E}[X]=\frac{\alpha }{\beta }$ Median No simple closed form No simple closed form Mode $(k-1)\theta \text{ for }k\ge 1$ $\frac{\alpha -1}{\beta }\text{ for }\alpha \ge 1$ Variance $\mathrm{Var}(X)=k{\theta }^2$ $\mathrm{Var}(X)=\frac{\alpha }{{\beta }^2}$ Skewness $\frac{2}{\sqrt{k}}$ $\frac{2}{\sqrt{\alpha }}$ Excess kurtosis $\frac{6}{k}$ $\frac{6}{\alpha }$ Entropy $\begin{array}{rl}k& +\ln \theta +\ln \mathrm{\Gamma }(k)\\ & +(1-k)\psi (k)\end{array}$ $\begin{array}{rl}\alpha & -\ln \beta +\ln \mathrm{\Gamma }(\alpha )\\ & +(1-\alpha )\psi (\alpha )\end{array}$ MGF $(1-\theta t{)}^{-k}\text{ for }t<\frac{1}{\theta }$ ${(1-\frac{t}{\beta })}^{-\alpha }\text{ for }t<\beta$ CF $(1-\theta it{)}^{-k}$ ${(1-\frac{it}{\beta })}^{-\alpha }$

Gamma Probability density function Probability density plots of gamma distributions Cumulative distribution function Cumulative distribution plots of gamma distributions Parameters k > 0 shape θ > 0 scale α > 0 shape β > 0 rate Support $x\in (0,\mathrm{\infty })$ $x\in (0,\mathrm{\infty })$ PDF $\frac{1}{\mathrm{\Gamma }(k){\theta }^k}{x}^{k-1}{e}^{-\frac{x}{\theta }}$ $\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{x}^{\alpha -1}{e}^{-\beta x}$ CDF $\frac{1}{\mathrm{\Gamma }(k)}\gamma (k,\frac{x}{\theta })$ $\frac{1}{\mathrm{\Gamma }(\alpha )}\gamma (\alpha ,\beta x)$ Mean $\mathrm{E}[X]=k\theta$ $\mathrm{E}[X]=\frac{\alpha }{\beta }$ Median No simple closed form No simple closed form Mode $(k-1)\theta \text{ for }k\ge 1$ $\frac{\alpha -1}{\beta }\text{ for }\alpha \ge 1$ Variance $\mathrm{Var}(X)=k{\theta }^2$ $\mathrm{Var}(X)=\frac{\alpha }{{\beta }^2}$ Skewness $\frac{2}{\sqrt{k}}$ $\frac{2}{\sqrt{\alpha }}$ Ex. kurtosis $\frac{6}{k}$ $\frac{6}{\alpha }$ Entropy $\begin{array}{rl}k& +\ln \theta +\ln \mathrm{\Gamma }(k)\\ & +(1-k)\psi (k)\end{array}$ $\begin{array}{rl}\alpha & -\ln \beta +\ln \mathrm{\Gamma }(\alpha )\\ & +(1-\alpha )\psi (\alpha )\end{array}$ MGF $(1-\theta t{)}^{-k}\text{ for }t<\frac{1}{\theta }$ ${(1-\frac{t}{\beta })}^{-\alpha }\text{ for }t<\beta$ CF $(1-\theta it{)}^{-k}$ ${(1-\frac{it}{\beta })}^{-\alpha }$