Information Technology Reference
In-Depth Information
11. Gumbel distribution.
PDF:
f
(
x
;
β
)=
β
e
−x/β
e
−e
−
x/β
;
x
∈
R
;
β>
0.
V
:
π
6
β
2
.
H
:ln
β
+
γ
+1.
Comment:
γ
is the Euler-Mascheroni constant.
12. Inverse Chi-square distribution.
PDF:
f
(
x
;
ν
)=
2
−
ν/
2
Γ
(
ν/
2)
x
−ν/
2
−
1
e
−
1
/
(2
x
)
;
x>
0;
ν>
0.
2
V
:
(
ν−
2)
2
(
ν−
4)
,ν>
4.
2
+ln(
2
Γ
(
2
))
−
(1 +
2
)
ψ
(
2
).
Comment: a special case of the gamma distribution.
ν
H
:
13. Inverse Gaussian distribution (also known as Wald distribution).
PDF:
f
(
x
;
λ, μ
)=
λ
2
πx
3
1
/
2
exp
−λ
(
x−μ
)
2
;
x>
0;
λ, μ >
0.
2
μ
2
x
V
:
μ
3
/λ
.
H
:
δ
δ
=
−
1
/
2
.
Comment:
H
(
V
) is USF only for fixed
m
. The entropy [158] uses the
derivative of the modified Bessel function of the second kind, which can
be computed as (
K
δ
(
x
))
δ
=
+
3
μ
e
μ
−
λ
+3
e
μ
K
δ
λ
1
2
+
2
ln
2
λ
1
−
2
(
K
δ−
1
(
x
)+
K
δ
+1
(
x
))
.
14. Laplace distribution.
PDF:
f
(
x
;
μ, σ
)=
√
2
σ
exp
;
x
√
2
1
−
σ
|
x
−
μ
|
∈
R
;
σ>
0.
V
:
σ
2
.
H
:ln(
σe
√
2).
15. Logarithmic distribution.
PDF:
f
(
x
;
a, b
)=
K
ln
x
with
K
=1
/
[
b
(ln
b −
1)
− a
(ln
a −
1)];
x ∈
[
a, b
];
a, b >
0
,a<b
.
V
:
K
9
[
b
3
(3 ln
b −
1)
− a
3
(3 ln
a −
1)]
− μ
2
K
4
[
b
2
(2 ln
b −
1)
−
with
μ
=
a
2
(2 ln
a
−
1)].
H
:
−
K
[
b
(ln
b
−
2)
−
a
(ln
a
−
2)+
Ei
(ln
b
)
−
Ei
(ln
a
)]
−
ln
K
,for
a, b >
1.
Comment:
Ei
is the exponential integral function.
16. Logistic distribution.
PDF:
f
(
x
;
μ, s
)=
e
−
(
x
−
μ
)
/s
s
(1+
e
−
(
x
−
μ
)
/s
)
2
;
x
∈
R
;
s>
0.
V
:
π
3
s
2
.
H
:ln
s
+2
.
17. Log-Logistic distribution.
PDF:
f
(
x
;
α, β
)=
(
β/α
)(
x/α
)
β
−
1
[1+(
x/α
)
β
]
2
;
x
≥
0
,α,β>
0.
V
:
α
2
2
π/β
sin
2
(
π/β
)
;
β>
2.
(
π/β
)
2
sin(2
π/β
)
−
ln
β
.
H
:2
−
