- Source: Type-2 Gumbel distribution
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 }
for
x
>
0
.
{\displaystyle \quad x>0~.}
For
0
<
a
≤
1
{\displaystyle \ 0
the mean is infinite. For
0
<
a
≤
2
{\displaystyle \ 0
the 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 [}X^{k}{\bigr ]}\ }
exist for
k
<
a
{\displaystyle \ k
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates
Given a random variate
U
{\displaystyle \ U\ }
drawn from the uniform distribution in the interval
(
0
,
1
)
,
{\displaystyle \ (0,1)\ ,}
then the variate
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\ }
and
b
.
{\displaystyle \ b~.}
This is obtained by applying the inverse transform sampling-method.
Related distributions
The special case
b
=
1
{\displaystyle \ 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
{\displaystyle \ k\ }
(as in the Weibull distribution) would yield a negative
a
{\displaystyle \ a\ }
and hence a negative probability density, which is not allowed.
Based on "Gumbel distribution". The GNU Scientific Library. type 002d2, used under GFDL.
See also
Extreme value theory
Gumbel distribution
Kata Kunci Pencarian:
- Amerika Serikat
- Type-2 Gumbel distribution
- Gumbel distribution
- Type 2
- List of probability distributions
- Generalized extreme value distribution
- Fréchet distribution
- List of statistics articles
- Weibull distribution
- Exponential distribution
- Generalized multivariate log-gamma distribution
