Template:Probability distribution Source: en.wikipedia.org/wiki/Template:Probability_distribution

Normal distribution
Normal distribution
	Probability density function The red curve is the standard normal distribution
	Cumulative distribution function
Notation
Parameters	= mean (location); = variance (squared scale)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
MAD
AAD
Skewness
Excess kurtosis
Entropy
MGF
CF
Fisher information
Kullback–Leibler divergence

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Example

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:

{{Infobox probability distribution
| name       = 
| type       = 
| pdf_image  = 
| cdf_image  = 
| notation   = 
| parameters = 
| support    = 
| pdf        = 
| cdf        = 
| quantile   = 
| mean       = 
| median     = 
| mode       = 
| variance   = 
| mad        =
| aad        = 
| skewness   = 
| kurtosis   = 
| entropy    = 
| cross_entropy = 
| mgf        = 
| cf         = 
| pgf        = 
| fisher     = 
| moments    = 
| KLdiv      = 
| likelihood = 
| JSDiv      = 
}}

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 captionn
|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 $σ 2$ for the normal distribution).
|support= — the support of the distribution, which may depend on the parameters. Specify this as <math>x \in some set</math> for continuous distributions, and as <math>k \in some set</math> for discrete distributions.
|pdf= — probability density function (or probability mass function), such as: <math>\frac{\Gamma(r+k)}{k!\Gamma(r)}p^r(1-p)^k</math>. Please exclude the function label, such as " $ƒ (x; μ, σ 2)$ ".
|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 median absolute deviation around the median.
|aad= — 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: <math>\left(\frac{p}{1-(1-p) e^t}\right)^r</math>.
|char=/|cf= — the characteristic function, such as: <math>\left(\frac{p}{1-(1-p) e^{it}}\right)^r</math>.
|pgf= - the probability-generating function.
|fisher= — the Fisher information matrix for the model.
|KLDiv= — the Kullback-Leibler divergence of the model
|likelihood= — the Likelihood function 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).

Tracking category

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

The above documentation is transcluded from Template:Infobox probability distribution/doc.
Editors can experiment in this template's sandbox and testcases pages.
Add categories to the /doc subpage. Subpages of this template.

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	$\sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(1/2)$
AAD	$\sigma {\sqrt {2/\pi }}$
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	${1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+2\ln {\sigma _{1} \over \sigma _{0}}\right\}$