Template:Infobox probability distribution/testcases

Normal distribution

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{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

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 <math>\mathcal{N}(\mu,\sigma^2)</math> Parameters <math>\mu\in\R</math> = mean (location) <math>\sigma^2>0</math> = variance (squared scale) Support <math>x\in\R</math> PDF <math>\frac{1}{\sqrt{2\pi\sigma^2

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 <math>\mathcal{N}(\mu,\sigma^2)</math> Parameters <math>\mu\in\R</math> = mean (location) <math>\sigma^2>0</math> = variance (squared scale) Support <math>x\in\R</math> PDF <math>\frac{1}{\sqrt{2\pi\sigma^2

e^{-\frac{(x - \mu)^2}{2 \sigma^2}}</math>

 | cdf        = <math>\frac{1}{2}\left[1 + \operatorname{erf}\left( \frac{x-\mu}{\sigma\sqrt{2}}\right)\right] </math>
 | quantile   = <math>\mu+\sigma\sqrt{2} \operatorname{erf}^{-1}(2p-1)</math>
 | mean       = <math>\mu</math>
 | median     = <math>\mu</math>
 | mode       = <math>\mu</math>
 | variance   = <math>\sigma^2</math>
 | mad        = <math>\sqrt{2/\pi}\sigma</math>
 | skewness   = <math>0</math>
 | kurtosis   = <math>0</math> 
 | entropy    = <math>\frac{1}{2} \log(2\pi e\sigma^2)</math>
 | mgf        = <math>\exp(\mu t + \sigma^2t^2/2)</math>
 | char       = <math>\exp(i\mu t - \sigma^2 t^2/2)</math>
 | fisher     = <math>\mathcal{I}(\mu,\sigma) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 2/\sigma^2\end{pmatrix}</math>

<math>\mathcal{I}(\mu,\sigma^2) =\begin {pmatrix} 1/\sigma^2 & 0 \\ 0 & 1/(2\sigma^4)\end{pmatrix}</math>

 | KLDiv = <math>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} \}</math>

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Binomial distribution

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{{Infobox probability distribution/sandbox}}

Binomial distribution Probability mass function Probability mass function for the binomial distribution Cumulative distribution function Cumulative distribution function for the binomial distribution Notation <math>B(n,p)</math> Parameters <math>n \in \{0, 1, 2, \ldots\}</math> – number of trials <math>p \in [0,1]</math> – success probability for each trial Support <math>k \in \{0, 1, \ldots, n\}</math> – number of successes PMF <math>\binom{n}{k} p^k (1-p)^{n-k}</math> CDF <math>I_{1-p}(n - k, 1 + k)</math> Mean <math>np</math> Median <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math> Mode <math>\lfloor (n + 1)p \rfloor</math> or <math>\lceil (n + 1)p \rceil - 1</math> Variance <math>np(1 - p)</math> Skewness <math>\frac{1-2p}{\sqrt{np(1-p)

Binomial distribution Probability mass function Probability mass function for the binomial distribution Cumulative distribution function Cumulative distribution function for the binomial distribution Notation <math>B(n,p)</math> Parameters <math>n \in \{0, 1, 2, \ldots\}</math> – number of trials <math>p \in [0,1]</math> – success probability for each trial Support <math>k \in \{0, 1, \ldots, n\}</math> – number of successes PMF <math>\binom{n}{k} p^k (1-p)^{n-k}</math> CDF <math>I_{1-p}(n - k, 1 + k)</math> Mean <math>np</math> Median <math>\lfloor np \rfloor</math> or <math>\lceil np \rceil</math> Mode <math>\lfloor (n + 1)p \rfloor</math> or <math>\lceil (n + 1)p \rceil - 1</math> Variance <math>np(1 - p)</math> Skewness <math>\frac{1-2p}{\sqrt{np(1-p)

</math>

 | kurtosis   = <math>\frac{1-6p(1-p)}{np(1-p)}</math>
 | entropy    = <math>\frac{1}{2} \log_2 \left( 2\pi enp(1-p) \right) + O \left( \frac{1}{n} \right)</math>
 in shannons. For nats, use the natural log in the log.
 | mgf        = <math>(1-p + pe^t)^n</math>
 | char       = <math>(1-p + pe^{it})^n</math>
 | pgf        = <math>G(z) = [(1-p) + pz]^n</math>
 | fisher     = <math> g_n(p) = \frac{n}{p(1-p)} </math>
(for fixed <math>n</math>)

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Geometric distribution

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{{Infobox probability distribution}}

{{Infobox probability distribution/sandbox}}

Geometric Probability mass function File:Geometric pmf.svg Cumulative distribution function File:Geometric cdf.svg Parameters <math>0< p < 1</math> success probability (real) Support k trials where <math>k \in \{1,2,3,\dots\}</math> PMF <math>(1 - p)^{k-1}p</math> CDF <math>1-(1 - p)^k</math> Mean <math>\frac{1}{p}</math> Median <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil</math> (not unique if <math>-1/\log_2(1-p)</math> is an integer) Mode <math>1</math> Variance <math>\frac{1-p}{p^2}</math> Skewness <math>\frac{2-p}{\sqrt{1-p

Geometric Probability mass function File:Geometric pmf.svg Cumulative distribution function File:Geometric cdf.svg Parameters <math>0< p < 1</math> success probability (real) Support k trials where <math>k \in \{1,2,3,\dots\}</math> PMF <math>(1 - p)^{k-1}p</math> CDF <math>1-(1 - p)^k</math> Mean <math>\frac{1}{p}</math> Median <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil</math> (not unique if <math>-1/\log_2(1-p)</math> is an integer) Mode <math>1</math> Variance <math>\frac{1-p}{p^2}</math> Skewness <math>\frac{2-p}{\sqrt{1-p

</math>

| kurtosis = <math>6+\frac{p^2}{1-p}</math> | entropy = <math>\tfrac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}</math> | mgf = <math>\frac{pe^t}{1-(1-p) e^t},</math>
for <math>t<-\ln(1-p)</math> | char = <math>\frac{pe^{it}}{1-(1-p)e^{it}}</math> | parameters2 = <math>0< p \leq 1</math> success probability (real) | support2 = k failures where <math>k \in \{0,1,2,3,\dots\}</math> | pdf2 = <math>(1 - p)^k p</math> | cdf2 = <math>1-(1 - p)^{k+1}</math> | mean2 = <math>\frac{1-p}{p}</math> | median2 = <math>\left\lceil \frac{-1}{\log_2(1-p)} \right\rceil - 1</math>
(not unique if <math>-1/\log_2(1-p)</math> is an integer) | mode2 = <math>0</math> | variance2 = <math>\frac{1-p}{p^2}</math> | skewness2 = <math>\frac{2-p}{\sqrt{1-p}}</math> | kurtosis2 = <math>6+\frac{p^2}{1-p}</math> | entropy2 = <math>\tfrac{-(1-p)\log_2 (1-p) - p \log_2 p}{p}</math> | mgf2 = <math>\frac{p}{1-(1-p)e^t}</math> | char2 = <math>\frac{p}{1-(1-p)e^{it}}</math> }}

Gamma distribution

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{{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 Support <math>x \in (0, \infty)</math> PDF <math>\frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta	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 Support <math>x \in (0, \infty)</math> PDF <math>\frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta

</math>

| cdf =<math>\frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{\theta}\right)</math> | mean =<math>\operatorname{E}[X] = k \theta </math> | median =No simple closed form | mode =<math>(k - 1)\theta \text{ for } k \geq 1</math> | variance =<math>\operatorname{Var}(X) = k \theta^2</math> | skewness =<math>\frac{2}{\sqrt{k}}</math> | kurtosis =<math>\frac{6}{k}</math> | entropy =<math>\begin{align}

                     k &+ \ln\theta + \ln\Gamma(k)\\
                       &+ (1 - k)\psi(k)
                   \end{align}</math>

| mgf =<math>(1 - \theta t)^{-k} \text{ for } t < \frac{1}{\theta}</math> | char =<math>(1 - \theta it)^{-k}</math> | parameters2 =

• α > 0 shape
• β > 0 rate

| support2 =<math>x \in (0, \infty)</math> | pdf2 =<math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x }</math> | cdf2 =<math>\frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x)</math> | mean2 =<math>\operatorname{E}[X] = \frac{\alpha}{\beta}</math> | median2 =No simple closed form | mode2 =<math>\frac{\alpha - 1}{\beta} \text{ for } \alpha \geq 1</math> | variance2 =<math>\operatorname{Var}(X) = \frac{\alpha}{\beta^2}</math> | skewness2 =<math>\frac{2}{\sqrt{\alpha}}</math> | kurtosis2 =<math>\frac{6}{\alpha}</math> | entropy2 =<math>\begin{align}

                     \alpha &- \ln \beta + \ln\Gamma(\alpha)\\
                            &+ (1 - \alpha)\psi(\alpha)
                   \end{align}</math>

| mgf2 =<math>\left(1 - \frac{t}{\beta}\right)^{-\alpha} \text{ for } t < \beta</math> | char2 =<math>\left(1 - \frac{it}{\beta}\right)^{-\alpha}</math> }}