Probability distribution
Type-2 Gumbel Parameters
a
∈
R
{\displaystyle \ a\in \mathbb {R} \ }
b
∈
R
{\displaystyle \ b\in \mathbb {R} \ }
Support
0
<
x
<
∞
{\displaystyle \ 0<x<\infty \ }
PDF
a
b
x
−
a
−
1
e
−
b
x
−
a
{\displaystyle \ a\ b\ x^{-a-1}\ e^{-b\ x^{-a}}\ }
CDF
e
−
b
x
−
a
{\displaystyle \ e^{-b\ x^{-a}}\ }
Quantile
(
−
log
e
(
p
)
b
)
−
1
a
{\displaystyle \ \left(-\ {\frac {\ \log _{e}\!\left(p\right)\ }{b}}\right)^{-{\frac {1}{a}}}\ }
Mean
b
1
a
Γ
(
1
−
1
a
)
{\displaystyle \ b^{\frac {1}{a}}\ \Gamma \!\left(\ 1-{\tfrac {\ 1\ }{a}}\ \right)\ }
Variance
b
2
a
Γ
(
1
−
1
a
)
(
1
−
Γ
(
1
−
1
a
)
)
{\displaystyle \ b^{\frac {2}{a}}\ \Gamma \!\left(1-{\tfrac {\ 1\ }{a}}\ \right){\Bigl (}1-\Gamma \!\left(1-{\tfrac {1}{a}}\right){\Bigr )}\ }
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)=a\ b\ x^{-a-1}\ e^{-b\ x^{-a}}\quad }
x
>
0
.
{\displaystyle \quad x>0~.}
For
0
<
a
≤
1
{\displaystyle \ 0<a\leq 1\ }
mean is infinite. For
0
<
a
≤
2
{\displaystyle \ 0<a\leq 2\ }
variance is infinite.
The cumulative distribution function is
F
(
x
|
a
,
b
)
=
e
−
b
x
−
a
.
{\displaystyle \ F(x|a,b)=e^{-b\ x^{-a}}~.}
The moments
E
[
X
k
]
{\displaystyle \ \mathbb {E} {\bigl }\ }
k
<
a
{\displaystyle \ k<a\ }
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates
Given a random variate
U
{\displaystyle \ U\ }
uniform distribution in the interval
(
0
,
1
)
,
{\displaystyle \ (0,1)\ ,}
X
=
(
−
ln
U
b
)
−
1
a
{\displaystyle X=\left(-{\frac {\ln U}{b}}\right)^{-{\frac {1}{a}}}\ }
has a Type-2 Gumbel distribution with parameter
a
{\displaystyle \ a\ }
b
.
{\displaystyle \ b~.}
inverse transform sampling -method.
Related distributions
Substituting
b
=
λ
−
k
{\displaystyle \ b=\lambda ^{-k}\ }
a
=
−
k
{\displaystyle \ a=-k\ }
Weibull distribution . Note, however, that a positive
k
{\displaystyle \ k\ }
a
{\displaystyle \ a\ }
Based on "Gumbel distribution" . The GNU Scientific Library . type 002d2,
See also
Category :
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