Type-2 Gumbel distribution explained

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 < inf ty .

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,&nbsp;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 =λ ^{-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.

See also

• Extreme value theory
• Gumbel distribution