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Please help improve this article or section by expanding it. Further information might be found on the or at requests for expansion. (May 2007) |
| Probability density function |
|
| Cumulative distribution function |
|
| Parameters | <math>a\!</math> (real) <math>b\!</math> shape (real) |
|---|---|
| Support | |
| Probability density function (pdf) | <math>a b x^{-a-1} \exp(-b x^{-a})\!</math> |
| Cumulative distribution function (cdf) | <math>\exp(-b x^{-a})\!</math> |
| Mean | |
| Median | |
| Mode | |
| Variance | |
| Skewness | |
| Excess kurtosis | |
| Entropy | |
| Moment-generating function (mgf) | |
| Characteristic function | |
In probability theory, the Type-2 Gumbel probability density function is
- <math>f(x|a,b) = a b x^{-a-1} \exp(-b x^{-a})\,</math>
for
- <math>0 < x < \infty</math>.
This implies that it similar to the Weibull distributions, substituting <math>b=\lambda^{-k}</math> and <math>a=-k</math>. Note however that a positive k (as in the Weibull distribution) would yield a negative a, which is not allowed here as it would yield a negative probability density. 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) = \exp(-b x^{-a})\,</math>
The moments <math> E[X^k] \,</math> exist for <math>k < a\,</math> The special case b = 1 yelds the Fréchet distribution
Based on gsl-ref_19.html#SEC309, used under GFDL.
