The Gumbel distribution is a continuous probability distribution with probability density function given by:

subject to the following conditions:

The Gumbel distribution is also known as the type I extreme value distribution.

The Gumbel variate with location parameter a and scale parameter b is related to the Exponential variate with scale parameter b according to the formula:  Gumbel(a,b) ~ a - log(Exponential(b))

Note that the Gumbel command is inert and should be used in combination with the RandomVariable command.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$X≔\mathrm{RandomVariable}\left(\mathrm{Gumbel}\left(a\,b\right)\right)\:$

$\mathrm{PDF}\left(X\,u\right)$

$\frac{{\ⅇ}^{-\frac{u-a}{b}}{\ⅇ}^{-{\ⅇ}^{-\frac{u-a}{b}}}}{b}$

$\mathrm{PDF}\left(X\,0.5\right)$

$\frac{{\ⅇ}^{-\frac{1.\left(0.5-1.a\right)}{b}}{\ⅇ}^{-1.{\ⅇ}^{-\frac{1.\left(0.5-1.a\right)}{b}}}}{b}$

$\mathrm{Mean}\left(X\right)$

$\mathrm{\gamma }b+a$

$\mathrm{Variance}\left(X\right)$

$\frac{b^2{\mathrm{\pi }}^2}{6}$

Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.

Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.