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 function and distribution function
(1)
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(2)
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The mean, variance, skewness, and kurtosis excess are
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(5)
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(6)
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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
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and distribution function
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The th raw moment is given by
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The first few central moments are
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The mean, variance, skewness, and kurtosis excess are therefore given by
(13)
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(15)
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(16)
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If are instead taken from a standard normal distribution, then the corresponding cumulative distribution is
(17)
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where is the normal distribution function. The probability distribution of is then
(19)
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(20)
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The mean and variance are expressible in closed form for small ,
(21)
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(22)
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and
(26)
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(28)
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(29)
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(30)
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No exact expression is known for or , but there is an equation connecting them
(31)
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