Template:Probability distribution/doc

Example

Normal distribution

Probability density function The red curve is the standard normal distribution
Cumulative distribution function
Notation	${𝒩}(\mu ,{\sigma }^2)$
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	$\frac{1}{2}[1+\mathrm{erf}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)]$
Quantile	$\mu +\sigma \sqrt{2}{\mathrm{erf}}^{-1}(2p-1)$
Mean	$\mu$
Median	$\mu$
Mode	$\mu$
Variance	${\sigma }^2$
MAD	$\sigma \sqrt{2}{\mathrm{erf}}^{-1}(1/2)$
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	${ℐ}(\mu ,\sigma )=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 2/{\sigma }^2\end{array}\right)$ ${ℐ}(\mu ,{\sigma }^2)=\left(\begin{array}{cc}1/{\sigma }^2& 0\\ 0& 1/(2{\sigma }^4)\end{array}\right)$
Kullback–Leibler divergence	$\frac{1}{2}\{{\left(\frac{{\sigma }_0}{{\sigma }_1}\right)}^2+\frac{({\mu }_1-{\mu }_0{)}^2}{{\sigma }_1^2}-1+2\ln \frac{{\sigma }_1}{{\sigma }_0}\}$

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      = 
| 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 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 ${𝒩}(\mu ,{\sigma }^2)$. 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 <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,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 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
• |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=, |aad=, |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)