The Gumbel distribution (after Emil Julius Gumbel ), the Fisher-Tippett distribution (after Ronald Aylmer Fisher ) or Extremal I distribution is a continuous probability distribution which, like the Rossi distribution and the Fréchet distribution, belongs to the extreme value distributions .
definition
Density function f (x) of the Gumbel distribution
A constant random variable is sufficient for a Gumbel distribution with scale parameters and position parameters if it has the probability density
X
{\ displaystyle X}
β
>
0
{\ displaystyle \ beta> 0}
μ
∈
R.
{\ displaystyle \ mu \ in \ mathbb {R}}
f
(
x
)
=
1
β
e
-
1
β
(
x
-
μ
)
e
-
e
-
1
β
(
x
-
μ
)
,
x
∈
R.
{\ displaystyle f (x) = {\ frac {1} {\ beta}} \ mathrm {e} ^ {- {\ frac {1} {\ beta}} (x- \ mu)} \ mathrm {e} ^ {- \ mathrm {e} ^ {- {\ frac {1} {\ beta}} (x- \ mu)}}, ~ x \ in \ mathbb {R}}
and with it the distribution function
F.
(
x
)
=
e
-
e
-
1
β
(
x
-
μ
)
,
x
∈
R.
{\ displaystyle F (x) = \ mathrm {e} ^ {- \ mathrm {e} ^ {- {\ frac {1} {\ beta}} (x- \ mu)}}, ~ x \ in \ mathbb {R}}
owns.
Standard case
If no parameters are given, the standard parameters and are meant. This gives the density
μ
=
0
{\ displaystyle \ mu = 0}
β
=
1
{\ displaystyle \ beta = 1}
f
(
x
)
=
e
-
x
e
-
e
-
x
,
x
∈
R.
{\ displaystyle f (x) = \ mathrm {e} ^ {- x} \ mathrm {e} ^ {- \ mathrm {e} ^ {- x}}, ~ x \ in \ mathbb {R}}
and the distribution function
F.
(
x
)
=
e
-
e
-
x
,
x
∈
R.
{\ displaystyle F (x) = \ mathrm {e} ^ {- \ mathrm {e} ^ {- x}}, ~ x \ in \ mathbb {R}}
The affine-linear transformations result in the entire class of distributions given above with the properties
X
↦
Y
: =
a
+
b
X
{\ displaystyle X \ mapsto Y: = a + bX}
F.
Y
(
x
)
=
F.
(
x
-
a
b
)
{\ displaystyle F_ {Y} (x) = F \ left ({\ frac {xa} {b}} \ right)}
f
Y
(
x
)
=
1
b
f
(
x
-
a
b
)
{\ displaystyle f_ {Y} (x) = {\ frac {1} {b}} f \ left ({\ frac {xa} {b}} \ right)}
E.
(
Y
)
=
b
E.
(
X
)
+
a
{\ displaystyle \ operatorname {E} (Y) = b \ operatorname {E} (X) + a}
Var
(
Y
)
=
b
2
Var
(
X
)
{\ displaystyle \ operatorname {Var} (Y) = b ^ {2} \ operatorname {Var} (X)}
.
properties
Expected value
The Gumbel distribution has the expected value
E.
(
X
)
=
μ
+
β
γ
{\ displaystyle \ operatorname {E} (X) = \ mu + \ beta \ gamma}
.
Where is the Euler-Mascheroni constant .
γ
≈
0.577
2
{\ displaystyle \ gamma \ approx 0 {,} 5772}
Variance
The variance of a Gumbel distribution is
Var
(
X
)
=
(
π
β
)
2
6th
{\ displaystyle \ operatorname {Var} (X) = {\ frac {(\ pi \ beta) ^ {2}} {6}}}
.
Standard deviation
The standard deviation of a Gumbel distribution is
σ
=
π
β
6th
{\ displaystyle \ sigma = {\ frac {\ pi \ beta} {\ sqrt {6}}}}
.
application
She will u. a. used in the following areas:
The Gumbel distribution is a typical distribution function for annual series. It can only be applied to series where the length of the measurement series corresponds to the sample size. Otherwise negative logarithms are obtained.
Relationship to other distributions
Relationship to extreme value distribution
The Gumbel distribution results from the extreme value distribution with the parameters , and .
a
=
μ
{\ displaystyle a = \ mu}
b
=
β
{\ displaystyle b = \ beta}
c
=
1
{\ displaystyle c = 1}
Web links
Discrete univariate distributions
Continuous univariate distributions
Multivariate distributions
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