Type-2 Gumbel distribution
Template:Short description Template:Probability distribution\ </math> |cdf = <math>\ e^{-b\ x^{-a}}\ </math> |quantile = <math>\ \left( -\ \frac{\ \log_e\!\left( p \right)\ }{ b } \right)^{-\frac{1}{a}}\ </math> |mean = <math>\ b^\frac{1}{a}\ \Gamma\!\left(\ 1 - \tfrac{\ 1\ }{ a }\ \right)\ </math> |median = |mode = |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 = |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 variatesEdit
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.
Related distributionsEdit
- 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.
Based on {{#invoke:citation/CS1|citation |CitationClass=web }} used under GFDL.