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 U drawn from the uniform distribution in the interval (0, 1), 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 b = 1 yields the Fréchet distribution. Substituting and 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 Continuous distributions