Skip to main content

Extreme Value Type I (Gumbel) Distribution

NIST/SEMATECH Section 1.3.6.6.16 Extreme Value Type I Distribution

PDF

Gumbel(0, 1) PDF -4 -2 0 2 4 6 8 x 0 0.1 0.2 0.3 0.4 f(x)

CDF

Gumbel(0, 1) CDF -4 -2 0 2 4 6 8 x 0 0.2 0.4 0.6 0.8 1 F(x)

Interactive Explorer

PDF

CDF

Formulas

Probability Density Function

f(x)=1βexp⁡ ⁣(−x−μβ−e−(x−μ)/β)f(x) = \frac{1}{\beta}\exp\!\left(-\frac{x-\mu}{\beta} - e^{-(x-\mu)/\beta}\right)

Cumulative Distribution Function

F(x)=e−e−(x−μ)/βF(x) = e^{-e^{-(x-\mu)/\beta}}

Properties

Mean

μ+β γE\mu + \beta\,\gamma_E

Variance

π2β26\frac{\pi^2\beta^2}{6}

Overview

The extreme value type I (Gumbel) distribution models the maximum or minimum of a large number of independent samples, with location μ\mu and scale β\beta. It is widely used in hydrology, meteorology, and structural engineering for modeling extreme events.

Related Distributions

Weibull Distribution Normal Distribution Double Exponential (Laplace) Distribution