Template:Infobox probability distribution/doc

Example

Normal distribution

Probability density function File:Normal Distribution PDF.svg The red curve is the standard normal distribution
Cumulative distribution function File:Normal Distribution CDF.svg
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}{\sigma\sqrt{2\pi

e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^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>\sigma\sqrt{2/\pi}</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>{ 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\}</math>

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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 <math>\mathcal{N}(\mu,\sigma^2)</math>. 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><math>x \in some set</math></syntaxhighlight> for continuous distributions, and as <syntaxhighlight lang="tex" inline><math>k \in some set</math></syntaxhighlight> for discrete distributions.
• |pdf= — probability density function (or probability mass function), such as: <syntaxhighlight lang="tex" inline><math>\frac{\Gamma(r+k)}{k!\Gamma(r)}p^r(1-p)^k</math></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 <math>F()</math> is the CDF and <math>Q()</math> is the quantile function, then <math>Q(F(x))=x</math>
• |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.
• |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><math>\left(\frac{p}{1-(1-p) e^t}\right)^r</math></syntaxhighlight>.
• |char=/|cf= — the characteristic function, such as: <syntaxhighlight lang="tex" inline><math>\left(\frac{p}{1-(1-p) e^{it}}\right)^r</math></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)