 Gumbel - Maple Help

Statistics[Distributions]

 Gumbel
 Gumbel distribution Calling Sequence Gumbel(a, b) GumbelDistribution(a, b) Parameters

 a - location parameter b - scale parameter Description

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

$f\left(t\right)=\frac{{ⅇ}^{-\frac{t-a}{b}}{ⅇ}^{-{ⅇ}^{-\frac{t-a}{b}}}}{b}$

 subject to the following conditions:

$a::\mathrm{real},0

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{Statistics}\right):$
 > $X≔\mathrm{RandomVariable}\left(\mathrm{Gumbel}\left(a,b\right)\right):$
 > $\mathrm{PDF}\left(X,u\right)$
 $\frac{{{ⅇ}}^{{-}\frac{{u}{-}{a}}{{b}}}{}{{ⅇ}}^{{-}{{ⅇ}}^{{-}\frac{{u}{-}{a}}{{b}}}}}{{b}}$ (1)
 > $\mathrm{PDF}\left(X,0.5\right)$
 $\frac{{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5}{-}{1.}{}{a}\right)}{{b}}}{}{{ⅇ}}^{{-}{1.}{}{{ⅇ}}^{{-}\frac{{1.}{}\left({0.5}{-}{1.}{}{a}\right)}{{b}}}}}{{b}}$ (2)
 > $\mathrm{Mean}\left(X\right)$
 ${\mathrm{\gamma }}{}{b}{+}{a}$ (3)
 > $\mathrm{Variance}\left(X\right)$
 $\frac{{{b}}^{{2}}{}{{\mathrm{\pi }}}^{{2}}}{{6}}$ (4) References

 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.