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Type-2 Gumbel distribution

Probability distribution

Summary

Probability distribution

\ b \in \Reals \ (scale) In probability theory, the Type-2 Gumbel probability density function is

:\ f(x|a,b) = a\ b\ x^{-a-1}\ e^{-b\ x^{-a}} \quad for \quad x 0 ~.

For \ 0 the mean is infinite. For \ 0 the variance is infinite.

The cumulative distribution function is

:\ F(x|a,b) = e^{ -b\ x^{-a} } ~.

The moments \ \mathbb{E}\bigl[ X^k \bigr]\ exist for \ k

The distribution is named after Emil Julius Gumbel (1891 – 1966).

Generating random variates

Given a random variate \ U\ drawn from the uniform distribution in the interval \ (0, 1)\ , then the variate

: X = \left(-\frac{\ln U}{b}\right)^{ -\frac{1}{a} }\

has a Type-2 Gumbel distribution with parameter \ a\ and \ b ~. This is obtained by applying the inverse transform sampling-method.

continuous-distributions
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