In probability theory, the Type-2 Gumbel probability density function is
for
For the mean is infinite. For the variance is infinite.
The cumulative distribution function is
The moments exist for
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates
Given a random variate drawn from the uniform distribution in the interval then the variate
has a Type-2 Gumbel distribution with parameter and This is obtained by applying the inverse transform sampling-method.
Related distributions
The special case yields the Fréchet distribution.
Substituting and yields the Weibull distribution. Note, however, that a positive (as in the Weibull distribution) would yield a negative and hence a negative probability density, which is not allowed.
If is Type-2 Gumbel-distributed with parameters and , then .
Based on used under GFDL.
See also Extreme value theory Gumbel distribution