Template:Infobox probability distribution/testcases

Normal distribution

Side by side comparison

{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

Normal Distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation $𝒩(\mu ,{\sigma }^2)$ Parameters $\mu \in ℝ$ = mean (location) ${\sigma }^2>0$ = variance (squared scale) Support $x\in ℝ$ 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 The red curve is the standard normal distribution Cumulative distribution function Notation $𝒩(\mu ,{\sigma }^2)$ Parameters $\mu \in ℝ$ = mean (location) ${\sigma }^2>0$ = variance (squared scale) Support $x\in ℝ$ 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

Side by side comparison

{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

Binomial distribution Probability mass function Cumulative distribution function 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 Error creating thumbnail: Cumulative distribution function 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

Side by side comparison

{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

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

Side by side comparison

{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

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 $\frac{1}{\Gamma (k){\theta }^k}{x}^{k-1}{e}^{-\frac{x}{\theta }}$ $\frac{{\beta }^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}{e}^{-\beta x}$ CDF $\frac{1}{\Gamma (k)}\gamma (k,\frac{x}{\theta })$ $\frac{1}{\Gamma (\alpha )}\gamma (\alpha ,\beta x)$ 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 $\begin{array}{cc}k& +\ln \theta +\ln \Gamma (k)\\ & +(1-k)\psi (k)\end{array}$ $\begin{array}{cc}\alpha & -\ln \beta +\ln \Gamma (\alpha )\\ & +(1-\alpha )\psi (\alpha )\end{array}$ 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 $\frac{1}{\Gamma (k){\theta }^k}{x}^{k-1}{e}^{-\frac{x}{\theta }}$ $\frac{{\beta }^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}{e}^{-\beta x}$ CDF $\frac{1}{\Gamma (k)}\gamma (k,\frac{x}{\theta })$ $\frac{1}{\Gamma (\alpha )}\gamma (\alpha ,\beta x)$ 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 $\begin{array}{cc}k& +\ln \theta +\ln \Gamma (k)\\ & +(1-k)\psi (k)\end{array}$ $\begin{array}{cc}\alpha & -\ln \beta +\ln \Gamma (\alpha )\\ & +(1-\alpha )\psi (\alpha )\end{array}$ MGF $(1-\theta t{)}^{-k}\text{ for }t<\frac{1}{\theta }$ ${(1-\frac{t}{\beta })}^{-\alpha }\text{ for }t<\beta$ CF $(1-\theta it{)}^{-k}$ ${(1-\frac{it}{\beta })}^{-\alpha }$