Type-2 Gumbel distribution

In probability theory, the Type-2 Gumbel probability density function is


 * $$f(x|a,b) = a b x^{-a-1} e^{-b x^{-a}}\,$$

for


 * $$0 < x < \infty$$.

For $$0<a\le 1$$ the mean is infinite. For $$0<a\le 2$$ the variance is infinite.

The cumulative distribution function is


 * $$F(x|a,b) = e^{-b x^{-a}}\,$$

The moments $$ E[X^k] \,$$ exist for $$k < a\,$$

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


 * $$X=(-\ln U/b)^{-1/a},$$

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

Related distributions

 * The special case b = 1 yields the Fréchet distribution.
 * Substituting $$b=\lambda^{-k}$$ and $$a=-k$$ yields the Weibull distribution. Note, however, that a positive k (as in the Weibull distribution) would yield a negative a and hence a negative probability density, which is not allowed.

Based on The GNU Scientific Library, used under GFDL.