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 ${\mathcal {N}}(\mu ,\sigma ^{2})$ 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 ${\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]$ Quantile $\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)$ 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 ${\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}$ ${\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}$ Kullback–Leibler divergence $D_{\text{KL}}({\mathcal {N}}_{0}\|{\mathcal {N}}_{1})={1 \over 2}\{(\sigma _{0}/\sigma _{1})^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+2\ln {\sigma _{1} \over \sigma _{0}}\}$

Normal Distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation ${\mathcal {N}}(\mu ,\sigma ^{2})$ 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 ${\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]$ Quantile $\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)$ 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 ${\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&0\\0&2/\sigma ^{2}\end{pmatrix}}$ ${\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}$ Kullback-Leibler divergence $D_{\text{KL}}({\mathcal {N}}_{0}\|{\mathcal {N}}_{1})={1 \over 2}\{(\sigma _{0}/\sigma _{1})^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+2\ln {\sigma _{1} \over \sigma _{0}}\}$

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 $n\in \{0,1,2,\ldots \}$ – number of trials $p\in [0,1]$ – success probability for each trial Support $k\in \{0,1,\ldots ,n\}$ – number of successes PMF ${\binom {n}{k}}p^{k}(1-p)^{n-k}$ 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 ${\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)$ in shannons. For nats, use the natural log in the log. MGF $(1-p+pe^{t})^{n}$ CF $(1-p+pe^{it})^{n}$ 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 Cumulative distribution function Notation $B(n,p)$ Parameters $n\in \{0,1,2,\ldots \}$ – number of trials $p\in [0,1]$ – success probability for each trial Support $k\in \{0,1,\ldots ,n\}$ – number of successes PMF ${\binom {n}{k}}p^{k}(1-p)^{n-k}$ 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 ${\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)$ in shannons. For nats, use the natural log in the log. MGF $(1-p+pe^{t})^{n}$ CF $(1-p+pe^{it})^{n}$ 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 Cumulative distribution function Parameters $0<p<1$ success probability (real) $0<p\leq 1$ 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 $\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil$ (not unique if $-1/\log _{2}(1-p)$ is an integer) $\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil -1$ (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 ${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$ ${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$ MGF ${\frac {pe^{t}}{1-(1-p)e^{t}}},$ for $t<-\ln(1-p)$ ${\frac {p}{1-(1-p)e^{t}}}$ CF ${\frac {pe^{it}}{1-(1-p)e^{it}}}$ ${\frac {p}{1-(1-p)e^{it}}}$

Geometric Probability mass function Cumulative distribution function Parameters $0<p<1$ success probability (real) $0<p\leq 1$ 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 $\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil$ (not unique if $-1/\log _{2}(1-p)$ is an integer) $\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil -1$ (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 ${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$ ${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$ MGF ${\frac {pe^{t}}{1-(1-p)e^{t}}},$ for $t<-\ln(1-p)$ ${\frac {p}{1-(1-p)e^{t}}}$ CF ${\frac {pe^{it}}{1-(1-p)e^{it}}}$ ${\frac {p}{1-(1-p)e^{it}}}$

Gamma distribution

Side by side comparison

{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

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

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