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 function and distribution function

			(1)
			(2)

The mean, variance, skewness, and kurtosis 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 distribution function

(8)

The th raw moment is given by

(9)

The first few central moments are

			(10)
			(11)
			(12)

The mean, variance, skewness, and kurtosis excess are therefore given by

			(13)
			(14)
			(15)
			(16)

If are instead taken from a standard normal distribution, then the corresponding cumulative distribution is

			(17)
			(18)

where is the normal distribution function. The probability distribution of is then

			(19)
			(20)

The mean and variance are expressible in closed form for small ,

			(21)
			(22)
			(23)
			(24)
			(25)

and

			(26)
			(27)
			(28)
			(29)
			(30)

No exact expression is known for or , but there is an equation connecting them

(31)

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