Type-2 Gumbel distribution

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Type-2 Gumbel
Probability density function
Cumulative distribution function
Parameters a\! (real)
b\! shape (real)
Support
Probability density function (pdf) a b x^{-a-1} \exp(-b x^{-a})\!
Cumulative distribution function (cdf) \exp(-b x^{-a})\!
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

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

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

for

0 < x < \infty.

This implies that it similar to the Weibull distributions, substituting b = λ − 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) = \exp(-b x^{-a})\,

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

The special case b = 1 yelds the Fréchet distribution

Based on gsl-ref_19.html#SEC309, used under GFDL.

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