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{{Documentation subpage}}
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{{Lua|Module:Infobox3cols|Module:InfoboxImage|Module:Check for unknown parameters}}{{TemplateStyles|Template:Infobox probability distribution/styles.css}}
{{Infobox probability distribution
| name = Normal distribution
| type = density
| pdf_image = Normal Distribution PDF.svg
| pdf_caption = The red curve is the ''standard normal distribution''
| cdf_image = Normal Distribution CDF.svg
| cdf_caption =
| notation = <math>\mathcal{N}(\mu,\sigma^2)</math>
| parameters = <math>\mu\in\R</math> = mean ([[___location parameter|___location]])<br /><math>\sigma^2>0</math> = variance (squared [[scale parameter|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}\,\operatorname{erf}^{-1}(1/2)</math>
| aad = <math>\sigma\sqrt{2/\pi}</math>
| skewness = <math>0</math>
| kurtosis = <math>0</math> <!-- DO NOT REPLACE THIS WITH THE OLD-STYLE KURTOSIS WHICH IS 3. -->
| 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>
}}
== 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">
{{Infobox probability distribution
| name =
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| mode =
| variance =
| mad =
| aad =
| skewness =
| kurtosis =
| entropy =
| cross_entropy =
| mgf =
| cf =
| pgf =
| fisher =
| moments =
| KLdiv =
| likelihood =
| JSDiv =
}}
</syntaxhighlight>
== Parameters ==
* {{para|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)
* {{para|type}} — possible values are
* {{para|pdf_image}} — probability density image-spec, such as: <code><nowiki>xxx.svg</nowiki></code>.
* {{para|pdf_caption}} — probability density image
* {{para|pdf_image_alt}} — [[WP:ALT|alternative text]] for the image in {{para|pdf_image}}
* {{para|cdf_image}} — cumulative distribution image-spec, such as: <code><nowiki>yyy.svg</nowiki></code>.
* {{para|cdf_caption}} — cumulative distribution image caption
* {{para|cdf_image_alt}} — [[WP:ALT|alternative text]] for the image in {{para|cdf_image}}
* {{para|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.
* {{para|parameters}} — parameters of the distribution family (such as {{mvar|μ}} and {{math|''σ''<sup>2</sup>}} for the normal distribution).
* {{para|support}} — the support of the distribution, which may depend on the parameters. Specify this as <
* {{para|pdf}} — probability density function (or probability mass function), such as: <
* {{para|cdf}} — cumulative distribution function, e.g.: <code><nowiki><math>I_p(r,k+1)\text{ where }I_p(x,y)</math> is the [[regularized incomplete beta function]]</nowiki></code>.
* {{para|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>
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* {{para|mode}} — the [[mode (statistics)|mode]].
* {{para|variance}} — [[variance]] of the distribution, or [[covariance matrix]] in multivariate case.
* {{para|mad}} — the [[median absolute deviation]] around the median.
* {{para|aad}} — the [[average absolute deviation|mean absolute deviation]] around the mean.
* {{para|skewness}} — the [[skewness]].
* {{para|kurtosis}} — the [[kurtosis excess]].
* {{para|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.
* {{para|cross_entropy}} — the [[cross-entropy]] of the model
* {{para|mgf}} — the [[moment-generating function]], for example: < * {{para|char}}
* {{para|pgf}} - the [[
* {{para|fisher}} — the [[Fisher information matrix]] for the model.
* {{para|KLDiv}} — the [[Kullback-Leibler divergence]] of the model
* {{para|likelihood}} — the [[Likelihood function]] of the model
* {{para|JSDiv}} — the [[Jensen-Shannon divergence]] of the model
* {{para|moments}} — formulas to use in [[Method of moments (statistics)|method of moments]] for the model.
* {{para|ES}} — the [[expected shortfall]] or CVaR for the model.
* {{para|bPOE}} — the buffered probability of exceedance for the model.
* {{para|intro}} — optional message which will be displayed before all other content in the infobox.
* {{para|marginleft}} — margin space left of infobox (default: 1em).
* {{para|box_width}} — width of the infobox (default: 325px).
{{para|parameters2}}, {{para|support2}}, {{para|pdf2}}, {{para|cdf2}}, {{para|mean2}}, {{para|median2}}, {{para|mode2}}, {{para|variance2}}, {{para|mad}}, {{para|aad}}, {{para|skewness2}}, {{para|kurtosis2}}, {{para|entropy2}}, {{para|mgf2}}, {{para|char2}}/{{para|cf2}}, {{para|moments2}}, {{para|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]].
▲== See also ==
== Tracking category ==
*
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