Type-2 Gumbel distribution

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Type-2 Gumbel
Parameters a\! (real)
b\! shape (real)
PDF  a b x^{-a-1} e^{-b x^{-a}}\!
CDF  e^{-b x^{-a}}\!

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

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


0 < x < \infty.

This implies that it is similar to the Weibull distributions, substituting b=\lambda^{-k} and a=-k. Note however that a positive k (as in the Weibull distribution) would yield a negative a, which is not allowed here as it would yield a negative probability density.

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 special case b = 1 yields the Fréchet distribution

Based on The GNU Scientific Library, used under GFDL.

See also