Gumbel distribution

The Gumbel distribution (after Emil Julius Gumbel ), the Fisher-Tippett distribution (after Ronald Aylmer Fisher ) or Extremal I distribution is a continuous probability distribution which, like the Rossi distribution and the Fréchet distribution, belongs to the extreme value distributions .

definition

Density function f (x) of the Gumbel distribution

A constant random variable is sufficient for a Gumbel distribution with scale parameters and position parameters if it has the probability density ${\displaystyle X}$ ${\displaystyle \beta >0}$ ${\displaystyle \mu \in \mathbb {R} }$

${\displaystyle f(x)={\frac {1}{\beta }}\mathrm {e} ^{-{\frac {1}{\beta }}(x-\mu )}\mathrm {e} ^{-\mathrm {e} ^{-{\frac {1}{\beta }}(x-\mu )}},~x\in \mathbb {R} }$

and with it the distribution function

${\displaystyle F(x)=\mathrm {e} ^{-\mathrm {e} ^{-{\frac {1}{\beta }}(x-\mu )}},~x\in \mathbb {R} }$

owns.

Standard case

If no parameters are given, the standard parameters and are meant. This gives the density ${\displaystyle \mu =0}$ ${\displaystyle \beta =1}$

${\displaystyle f(x)=\mathrm {e} ^{-x}\mathrm {e} ^{-\mathrm {e} ^{-x}},~x\in \mathbb {R} }$

and the distribution function

${\displaystyle F(x)=\mathrm {e} ^{-\mathrm {e} ^{-x}},~x\in \mathbb {R} }$

The affine-linear transformations result in the entire class of distributions given above with the properties ${\displaystyle X\mapsto Y:=a+bX}$

${\displaystyle F_{Y}(x)=F\left({\frac {x-a}{b}}\right)}$

${\displaystyle f_{Y}(x)={\frac {1}{b}}f\left({\frac {x-a}{b}}\right)}$

${\displaystyle \operatorname {E} (Y)=b\operatorname {E} (X)+a}$

${\displaystyle \operatorname {Var} (Y)=b^{2}\operatorname {Var} (X)}$.

properties

Expected value

The Gumbel distribution has the expected value

${\displaystyle \operatorname {E} (X)=\mu +\beta \gamma }$.

Where is the Euler-Mascheroni constant . ${\displaystyle \gamma \approx 0{,}5772}$

Variance

The variance of a Gumbel distribution is

${\displaystyle \operatorname {Var} (X)={\frac {(\pi \beta )^{2}}{6}}}$.

Standard deviation

The standard deviation of a Gumbel distribution is

${\displaystyle \sigma ={\frac {\pi \beta }{\sqrt {6}}}}$.

application

She will u. a. used in the following areas:

• Water management for extreme events such as floods and dry seasons
• Traffic planning
• Meteorology (weather forecast)
• Hydrology

The Gumbel distribution is a typical distribution function for annual series. It can only be applied to series where the length of the measurement series corresponds to the sample size. Otherwise negative logarithms are obtained.

Relationship to other distributions

Relationship to extreme value distribution

The Gumbel distribution results from the extreme value distribution with the parameters , and . ${\displaystyle a=\mu }$ ${\displaystyle b=\beta }$ ${\displaystyle c=1}$

Web links

• Eric W. Weisstein: Extreme Value Distribution on MathWorld .