Information Technology Reference
In-Depth Information
18. Lognorm distribution.
PDF:
f
(
x
;
μ, σ
)=
xσ
√
2
π
e
−
(ln
x
−
μ
)
2
1
;
x>
0
,σ>
0.
2
σ
2
V
:(
e
σ
2
1)
e
2
μ
+
σ
2
;
β>
2.
−
1+ln(2
πσ
2
)
2
H
:
+
μ
.
19. Maxwell-Boltzmann distribution.
PDF:
f
(
x
;
a
)=
π
x
2
e
−
x
2
/
(2
a
2
)
;
x
≥
0
,a>
0.
a
3
V
:
a
2
(3
π−
8
π
.
H
: (no known formula)
Comment: a special case of the Gamma distribution.
20. Pareto distribution (also known as Power law distribution).
PDF:
f
(
x
;
α, x
m
)=
αx
m
x
α
+1
;
x>x
m
,α,x
m
>
0.
αx
2
m
V
:
2)
,α>
2.
(
α
−
1)
2
(
α
−
H
:ln
x
m
1
1.
Comment: a special case of Pearson Type XI distribution
−
α
−
21. Pearson Type II distribution.
PDF:
f
(
x
;
a, m
)=
a
√
πΓ
(
m
+1)
1
m
Γ
(
m
+
2
)
x
2
a
2
−
;
x
∈
[
−
a, a
]
,a>
0
,m>
−
1.
a
2
Γ
(
m
+
2
)
Γ
(
m
+
2
)
1
2
V
:
;
m>
5
/
2
.
Γ
(
m
+
2
)
4
Γ
(
m
+
2
)
ma
1
−
2
m
(ln 2
−
1)
a
√
πΓ
(
m
+1)
H
:
−
ln
a
√
πΓ
(
m
+1)
−
;
m>
3
/
2
.
Comment:
H
(
V
) is USF only for fixed
m
.
22. Pearson Type IV distribution.
PDF:
f
(
x
;
k, a, m
)=
K
(1 + (
x/a
)
2
)
−m
exp(
−
k
atan(
x/a
));
x
∈
R
;
a, k >
0
,m>
1
/
2.
V
:
a
2
r
2
(
r−
1)
(
r
2
+1)for
k
=1;
r
=2
m
−
2
,m>
3
/
2.
I
(
r
)
F
(
r
)
+ln
a
+ln
F
(
r
).
with
F
(
r
)=
π/
2
H
:
−
I
(
r
)=
π/
2
−π/
2
e
−θ
cos
r
(
θ
)
dθ
and
−π/
2
e
−θ
cos
r
(
θ
)((
r
+
θ
)
dθ
.
23. Pearson Type V distribution (also known as Inverse-Gamma distri-
bution).
PDF:
f
(
x
;
α, β
)=
β
α
2) ln cos
θ
−
1
x
α
+1
exp(
−
β/x
)
/Γ
(
α
);
x>
0;
α, β >
0.
V
:
β
2
/
[(
α
1)
2
(
α
−
−
2)].
H
:
α
+ln(
βΓ
(
α
))
−
(1 +
α
)
ψ
(
α
).
24. Pearson Type VII distribution.
PDF:
f
(
x
;
a, m
)=
K
1+
x
2
a
2
−m
with
K
=
a
√
π
Γ
(
m−
1
/
2)
Γ
(
m
)
;
x
∈
R
;
a
=
0
,m>
1
/
2.
