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### Extreme Value Distribution

There are essentially three types of Fisher-Tippett extreme value distributions. The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions. These are distributions of an extreme order statistic for a distribution of elements .The Fisher-Tippett distribution corresponding to a maximum extreme value distribution (i.e., the distribution of the maximum ), sometimes known as the log-Weibull distribution, with location parameter and scale parameter is implemented in the Wolfram Language as ExtremeValueDistribution[alpha, beta].It has probability density functionand distribution function(1)(2)The moments can be computed directly by defining(3)(4)(5)Then the raw moments are(6)(7)(8)(9)(10)(11)where are Euler-Mascheroni integrals. Plugging in the Euler-Mascheroni integrals gives(12)(13)(14)(15)(16)where is the Euler-Mascheroni constant and is Apéry's..

### Gumbel Distribution

There are essentially three types of Fisher-Tippett extreme value distributions. The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions. These are distributions of an extreme order statistic for a distribution of elements . In this work, the term "Gumbel distribution" is used to refer to the distribution corresponding to a minimum extreme value distribution (i.e., the distribution of the minimum ).The Gumbel distribution with location parameter and scale parameter is implemented in the Wolfram Language as GumbelDistribution[alpha, beta].It has probability density functionand distribution function(1)(2)The mean, variance, skewness,and kurtosis excess are(3)(4)(5)(6)where is the Euler-Mascheroni constant and is Apéry's constant.The distribution of taken from a continuous uniform distribution over the unit interval has probability function(7)and..

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