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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