Probability distribution
Type-2 Gumbel 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}}\!} Quantile ( − ln ( p ) b ) − 1 a {\displaystyle \left(-{\frac {\ln(p)}{b}}\right)^{-{\frac {1}{a}}}} Mean b 1 / a Γ ( 1 − 1 / a ) {\displaystyle b^{1/a}\Gamma (1-1/a)\!} Variance b 2 / a ( Γ ( 1 − 1 / a ) − Γ ( 1 − 1 / a ) 2 ) {\displaystyle b^{2/a}(\Gamma (1-1/a)-{\Gamma (1-1/a)}^{2})\!}
In probability theory , the Type-2 Gumbel probability density function is
f ( x | a , b ) = a b x − a − 1 e − b x − a {\displaystyle f(x|a,b)=abx^{-a-1}e^{-bx^{-a}}\,} for
0 < x < ∞ {\displaystyle 0<x<\infty } . For 0 < a ≤ 1 {\displaystyle 0<a\leq 1} the mean is infinite. For 0 < a ≤ 2 {\displaystyle 0<a\leq 2} the variance is infinite.
The cumulative distribution function is
F ( x | a , b ) = e − b x − a {\displaystyle F(x|a,b)=e^{-bx^{-a}}\,} The moments E [ X k ] {\displaystyle E[X^{k}]\,} exist for k < a {\displaystyle k<a\,}
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates [ edit ] Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
X = ( − ln U / b ) − 1 / a , {\displaystyle X=(-\ln U/b)^{-1/a},} has a Type-2 Gumbel distribution with parameter a {\displaystyle a} and b {\displaystyle b} . This is obtained by applying the inverse transform sampling -method.
The special case b = 1 yields the Fréchet distribution . Substituting b = λ − k {\displaystyle b=\lambda ^{-k}} and a = − k {\displaystyle a=-k} 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.
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families