Type-2 Gumbel distribution
| Parameters |
[math]\displaystyle{ a\! }[/math] (real) [math]\displaystyle{ b\! }[/math] shape (real) | ||
|---|---|---|---|
| [math]\displaystyle{ a b x^{-a-1} e^{-b x^{-a}}\! }[/math] | |||
| CDF | [math]\displaystyle{ e^{-b x^{-a}}\! }[/math] | ||
| Mean | [math]\displaystyle{ b^{1/a}\Gamma(1-1/a)\! }[/math] | ||
| Variance | [math]\displaystyle{ b^{2/a}(\Gamma(1-1/a)-{\Gamma(1-1/a)}^2)\! }[/math] | ||
In probability theory, the Type-2 Gumbel probability density function is
- [math]\displaystyle{ f(x|a,b) = a b x^{-a-1} e^{-b x^{-a}}\, }[/math]
for
- [math]\displaystyle{ 0 \lt x \lt \infty }[/math].
For [math]\displaystyle{ 0\lt a\le 1 }[/math] the mean is infinite. For [math]\displaystyle{ 0\lt a\le 2 }[/math] the variance is infinite.
The cumulative distribution function is
- [math]\displaystyle{ F(x|a,b) = e^{-b x^{-a}}\, }[/math]
The moments [math]\displaystyle{ E[X^k] \, }[/math] exist for [math]\displaystyle{ k \lt a\, }[/math]
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
- [math]\displaystyle{ X=(-\ln U/b)^{-1/a}, }[/math]
has a Type-2 Gumbel distribution with parameter [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math]. This is obtained by applying the inverse transform sampling-method.
Related distributions
- The special case b = 1 yields the Fréchet distribution.
- Substituting [math]\displaystyle{ b=\lambda^{-k} }[/math] and [math]\displaystyle{ a=-k }[/math] yields the Weibull distribution. Note, however, that a positive k (as in the Weibull distribution) would yield a negative a and hence a negative probability density, which is not allowed.
Based on The GNU Scientific Library, used under GFDL.
See also
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