Type 2 Gumbel distribution - Alchetron, the free social encyclopedia


Parameters a {\displaystyle a\!} (real) b {\displaystyle b\!} shape (real) PDF a b x − a − 1 e − b x − a {\displaystyle abx^{-a-1}e^{-bx^{-a}}\!} CDF e − b x − a {\displaystyle e^{-bx^{-a}}\!}

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

f ( x | a , b ) = a b x − a − 1 e − b x − a

for

0 < x < ∞ .

This implies that it is similar to the Weibull distributions, substituting b = λ − k and a = − k . 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 0 < a ≤ 1 the mean is infinite. For 0 < a ≤ 2 the variance is infinite.

The cumulative distribution function is

F ( x | a , b ) = e − b x − a

The moments E [ X k ] exist for k < a

The special case b = 1 yields the Fréchet distribution

Based on The GNU Scientific Library, used under GFDL.

References

Type-2 Gumbel distribution Wikipedia

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