Editing Gumbel distribution (section)

{{Short description|Particular case of the generalized extreme value distribution}}
{{Probability distribution 
 |name       =Gumbel
 |type       =density
 |pdf_image  =[[File:Gumbel-Density.svg|325px|Probability distribution function]]
 |cdf_image  =[[File:Gumbel-Cumulative.svg|325px|Cumulative distribution function]]
 |parameters =<math>\mu,</math> [[location parameter|location]] ([[real numbers|real]])<br /><math>\beta>0,</math> [[scale parameter|scale]] (real)
 |support    =<math>x\in\mathbb{R}</math>
 |pdf        =<math>\frac{1}{\beta}e^{-(z+e^{-z})}</math><br /> where <math>z=\frac{x-\mu}{\beta}</math>
 |cdf        =<math>e^{-e^{-(x-\mu)/\beta}}</math>
 |quantile   =<math>\mu-\beta\ln(-\ln(p))</math>
 |mean       =<math>\mu + \beta\gamma</math> <br> where <math>\gamma</math> is the [[Euler&ndash;Mascheroni constant]]
 |median     =<math>\mu - \beta\ln(\ln 2)</math>
 |mode       =<math>\mu</math>
 |variance   =<math>\frac{\pi^2}{6}\beta^2</math>
 |skewness   =<math>\frac{12\sqrt{6}\,\zeta(3)}{\pi^3} \approx 1.14</math>
 |kurtosis   =<math>\frac{12}{5}</math>
 |entropy    =<math>\ln(\beta)+\gamma+1</math>
 |mgf        =<math>\Gamma(1-\beta t) e^{\mu t}</math>
 |char       =<math>\Gamma(1-i\beta t) e^{i\mu t}</math>
|notation=<math>\text{Gumbel}(\mu, \beta)</math>}}
In [[probability theory]] and [[statistics]], the '''Gumbel distribution''' (also known as the '''type-I [[generalized extreme value distribution]]''') is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.

This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to [[extreme value theory]], which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type.{{efn|This article uses the Gumbel distribution to model the distribution of the maximum value''. ''To model the minimum  value, use the negative of the original values.}}

The Gumbel distribution is a particular case of the [[generalized extreme value distribution]] (also known as the Fisher–Tippett distribution). It is also known as the ''log-[[Weibull distribution]]'' and the ''double exponential distribution'' (a term that is alternatively sometimes used to refer to the [[Laplace distribution]]). It is related to the [[Gompertz distribution]]: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained.

In the [[latent variable]] formulation of the [[multinomial logit]] model — common in [[discrete choice]] theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributed [[random variable]]s has a [[logistic distribution]].

The Gumbel distribution is named after [[Emil Julius Gumbel]] (1891&ndash;1966), based on his original papers describing the distribution.<ref>{{Citation |url= http://archive.numdam.org/article/AIHP_1935__5_2_115_0.pdf |title= Les valeurs extrêmes des distributions statistiques |last= Gumbel |first= E.J. |journal=  Annales de l'Institut Henri Poincaré |volume= 5 |year=  1935 |pages= 115–158 |issue= 2 |author-link= Emil Julius Gumbel}}</ref><ref>Gumbel E.J. (1941). "The return period of flood flows". The Annals of Mathematical Statistics, 12, 163–190.</ref>