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.

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 variables has a logistic distribution.

The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based on his original papers describing the distribution.

with cumulative distribution function and probability density function In this case the mode is 0, the median is

The cumulants, for n > 1, are given by The mode is μ, while the median is

are iid Gumbel random variables with parameters

is also a Gumbel random variable with parameters

is necessarily Gumbel distributed with scale parameter

(actually it suffices to consider just two distinct values of k>1 which are coprime).

Theory related to the generalized multivariate log-gamma distribution provides a multivariate version of the Gumbel distribution.

Gumbel has shown that the maximum value (or last order statistic) in a sample of random variables following an exponential distribution minus the natural logarithm of the sample size [7] approaches the Gumbel distribution as the sample size increases.

So the cumulative distribution of the maximum value

In hydrology, therefore, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes,[3] and also to describe droughts.

[9] Gumbel has also shown that the estimator r⁄(n+1) for the probability of an event — where r is the rank number of the observed value in the data series and n is the total number of observations — is an unbiased estimator of the cumulative probability around the mode of the distribution.

In number theory, the Gumbel distribution approximates the number of terms in a random partition of an integer[10] as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations.

[11] It appears in the coupon collector's problem.

In machine learning, the Gumbel distribution is sometimes employed to generate samples from the categorical distribution.

be independent samples of Gumbel(0, 1), then by routine integration,

Related equations include:[13] Since the quantile function (inverse cumulative distribution function),

is drawn from the uniform distribution on the interval

In pre-software times probability paper was used to picture the Gumbel distribution (see illustration).

The paper is based on linearization of the cumulative distribution function

: In the paper the horizontal axis is constructed at a double log scale.

-variable on the vertical axis, the distribution is represented by a straight line with a slope 1

When distribution fitting software like CumFreq became available, the task of plotting the distribution was made easier.