Template:Infobox probability distribution

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Example

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	${\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}$
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	$\sigma {\sqrt {2/\pi }}$
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	${1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+2\ln {\sigma _{1} \over \sigma _{0}}\right\}$

Usage

The Template:Infobox probability distribution generates a right-hand side infobox, based on the specified parameters. To use this template, copy the following code in your article and fill in as appropriate: <syntaxhighlight lang="wikitext">

</syntaxhighlight>

Parameters

• |name= — Name at the top of the infobox; should be the name of the distribution without the word "distribution" in it, e.g. "Normal", "Exponential" (optional)
• |type= — possible values are "discrete" (or "mass"), "continuous" (or "density"), and "multivariate"
• |pdf_image= — probability density image-spec, such as: xxx.svg.
• |pdf_caption= — probability density image caption
• |pdf_image_alt= — alternative text for the image in |pdf_image=
• |cdf_image= — cumulative distribution image-spec, such as: yyy.svg.
• |cdf_caption= — cumulative distribution image caption
• |cdf_image_alt= — alternative text for the image in |cdf_image=
• |notation= — typical designation for this distribution, for example ${\mathcal {N}}(\mu ,\sigma ^{2})$. The notation should include all the distribution parameters explained in the next cell.
• |parameters= — parameters of the distribution family (such as μ and σ² for the normal distribution).
• |support= — the support of the distribution, which may depend on the parameters. Specify this as <syntaxhighlight lang="tex" inline>$x\in someset$</syntaxhighlight> for continuous distributions, and as <syntaxhighlight lang="tex" inline>$k\in someset$</syntaxhighlight> for discrete distributions.
• |pdf= — probability density function (or probability mass function), such as: <syntaxhighlight lang="tex" inline>${\frac {\Gamma (r+k)}{k!\Gamma (r)}}p^{r}(1-p)^{k}$</syntaxhighlight>. Please exclude the function label, such as "ƒ(x; μ,σ²)".
• |cdf= — cumulative distribution function, e.g.: <math>I_p(r,k+1)\text{ where }I_p(x,y)</math> is the [[regularized incomplete beta function]].
• |quantile= — quantile function (or inverse cumulative distribution function). If $F()$ is the CDF and $Q()$ is the quantile function, then $Q(F(x))=x$
• |mean= — the mean, or expected value.
• |median= — the median, only for univariate distributions.
• |mode= — the mode.
• |variance= — variance of the distribution, or covariance matrix in multivariate case.
• |mad= — the mean absolute deviation around the mean.
• |skewness= — the skewness.
• |kurtosis= — the kurtosis excess.
• |entropy= — the differential information entropy, preferably expressed in unspecified units using base-unspecific log(.) rather than base-specific ln(.) which yields entropy in units of nats only.
• |cross_entropy= — the cross-entropy of the model
• |mgf= — the moment-generating function, for example: <syntaxhighlight lang="tex" inline>$\left({\frac {p}{1-(1-p)e^{t}}}\right)^{r}$</syntaxhighlight>.
• |char=/|cf= — the characteristic function, such as: <syntaxhighlight lang="tex" inline>$\left({\frac {p}{1-(1-p)e^{it}}}\right)^{r}$</syntaxhighlight>.
• |pgf= - the Probability-generating function.
• |fisher= — the Fisher information matrix for the model.
• |KLDiv= — the Kullback-Leibler divergence of the model
• |JSDiv= — the Jensen-Shannon divergence of the model
• |moments= — formulas to use in Method of moments for the model.
• |ES= — the Expected shortfall or CVaR for the model.
• |bPOE= — the Buffered Probability of Exceedance for the model.
• |intro= — optional message which will be displayed before all other content in the infobox.
• |marginleft= — margin space left of infobox (default: 1em).
• |box_width= — width of the infobox (default: 325px).

|parameters2=, |support2=, |pdf2=, |cdf2=, |mean2=, |median2=, |mode2=, |variance2=, |mad=, |skewness2=, |kurtosis2=, |entropy2=, |mgf2=, |char2=/|cf2=, |moments2=, |fisher2= are the same as their counterparts above. They should be used when the distribution needs two sets to describe it, e.g. Gamma distribution.

Tracking category

• Category:Pages using infobox probability distribution with unknown parameters (0)

The above documentation is transcluded from Template:Infobox probability distribution/doc. (edit | history)
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