\ </math>

|cdf = <math>\ e^{-b\ x^{-a\ </math>

|quantile = <math>\ \left( -\ \frac{\ \ln\!\left( p \right)\ }{ b } \right)^{-\frac{1}{a\ </math>

|mean = <math>\ b^\frac{1}{a}\ \Gamma\!\left(\ 1 - \tfrac{\ 1\ }{ a }\ \right)\ </math>

|median = <math>\ \left(\frac{\ln\left(2\right)}{b}\right)^{- {1 \over a </math>

|mode = <math>\ \left({ab \over a+1}\right)^{1 \over a} </math>

|variance = <math>\ b^\frac{2}{a}\ \Gamma\!\left( 1 - \tfrac{\ 1\ }{ a }\ \right) \Bigl( 1 - \Gamma\!\left( 1-\tfrac{1}{a}\right) \Bigr)\ </math>

|skewness =

|kurtosis =

|entropy = <math>\ \gamma\left(1+\frac{1}{a}\right)+\ln\left(\frac{b^{\frac{1}{a}{a}\right)+1</math>

|mgf =

|char =

In probability theory, the Type-2 Gumbel probability density function is

:<math>\ f(x|a,b) = a\ b\ x^{-a-1}\ e^{-b\ x^{-a \quad </math> for <math>\quad x > 0 ~.</math>

For <math>\ 0 < a \le 1\ </math> the mean is infinite. For <math>\ 0 < a \le 2\ </math> the variance is infinite.

The cumulative distribution function is

:<math>\ F(x|a,b) = e^{ -b\ x^{-a} } ~.</math>

The moments <math>\ \mathbb{E}\bigl[ X^k \bigr]\ </math> exist for <math>\ k < a\ </math>

The distribution is named after Emil Julius Gumbel (1891 – 1966).

Generating random variates

Given a random variate <math>\ U\ </math> drawn from the uniform distribution in the interval <math>\ (0, 1)\ ,</math> then the variate

:<math> X = \left(-\frac{\ln U}{b}\right)^{ -\frac{1}{a} }\ </math>

has a Type-2 Gumbel distribution with parameter <math>\ a\ </math> and <math>\ b ~.</math> This is obtained by applying the inverse transform sampling-method.

  • The special case <math>\ b = 1\ </math> yields the Fréchet distribution.
  • Substituting <math>\ b = \lambda^{-k}\ </math> and <math>\ a = -k\ </math> yields the Weibull distribution. Note, however, that a positive <math>\ k\ </math> (as in the Weibull distribution) would yield a negative <math>\ a\ </math> and hence a negative probability density, which is not allowed.
  • If <math> X </math> is Type-2 Gumbel-distributed with parameters <math>a</math> and <math>b</math>, then <math>X^{-1} \sim \mathrm{Weibull}(a,b^{-1/a})</math>.

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Based on used under GFDL.

See also

  • Extreme value theory
  • Gumbel distribution