Gumbel distribution

The cumulative distribution function of the Gumbel distribution is

${\displaystyle F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}.\,}$

The mode is μ, while the median is ${\displaystyle \mu -\beta \ln \left(\ln 2\right),}$ and the mean is given by

${\displaystyle \operatorname {E} (X)=\mu +\gamma \beta }$,

where ${\displaystyle \gamma }$ is the Euler-Mascheroni constant.

The standard deviation ${\displaystyle \sigma }$ is ${\displaystyle \beta \pi /{\sqrt {6}}}$ hence ${\displaystyle \beta =\sigma {\sqrt {6}}/\pi \approx 0.78\sigma .}$ [3]

At the mode, where ${\displaystyle x=\mu }$, the value of ${\displaystyle F(x;\mu ,\beta )}$ becomes ${\displaystyle e^{-1}\approx 0.37}$, irrespective of the value of ${\displaystyle \beta .}$

• If ${\displaystyle X}$ has a Gumbel distribution, then the conditional distribution of Y=−X given that Y is positive, or equivalently given that X is negative, has a Gompertz distribution. The cdf G of Y is related to F, the cdf of X, by the formula ${\displaystyle G(y)=P(Y\leq y)=P(X\geq -y|X\leq 0)=(F(0)-F(-y))/F(0)}$ for y>0. Consequently, the densities are related by ${\displaystyle g(y)=f(-y)/F(0)}$: the Gompertz density is proportional to a reflected Gumbel density, restricted to the positive half-line.[4]
• If X is an exponentially distributed variable with mean 1, then −log(X) has a standard Gumbel distribution.
• If ${\displaystyle X\sim \mathrm {Gumbel} (\alpha _{X},\beta )}$ and ${\displaystyle Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )}$ then ${\displaystyle X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,}$ (see Logistic distribution).
• If ${\displaystyle X}$ and ${\displaystyle Y\sim \mathrm {Gumbel} (\alpha ,\beta )}$ then ${\displaystyle X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )\,}$. Note that ${\displaystyle E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)}$.

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

Distribution fitting with confidence band of a cumulative Gumbel distribution to maximum one-day October rainfalls.[5]

Gumbel has shown that the maximum value (or last order statistic) in a sample of a random variable following an exponential distribution minus natural logarithm of the sample size [6] approaches the Gumbel distribution closer with increasing sample size.[7]

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.[8]

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. Therefore, this estimator is often used as a plotting position.

In number theory, the Gumbel distribution approximates the number of terms in a random partition of an integer[9] as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations.[10]

In machine learning, the Gumbel distribution is sometimes employed to generate samples from the categorical distribution.[11]

• Type-1 Gumbel distribution
• Type-2 Gumbel distribution
• Extreme value theory
• Generalized extreme value distribution
• Fisher–Tippett–Gnedenko theorem
• Emil Julius Gumbel