CODE 3926
Some Commonly Used Distributions
1.
Uniform
)
,
(
~
b
a
U
Y
2
]
[
b
a
Y
E
+
=
12
)
(
]
[
2
a
b
Y
V
−
=
a
b
b
a
y
f
−
=
1
)
,

(
b
y
a
≤
≤
2.
Beta
)
,
(
~
β
α
Be
Y
βα
+
=
]
[
Y
E
)
1
(
)
(
]
[
2
+
+
+
=
αβ
Y
V
1
1
)
1
(
)
,
(
1
)
,

(
−
−
−
=
y
y
B
y
f
0
;
0
;
1
0
>
>
≤
≤
y
3.
Normal
)
,
(
~
2
σθ
N
Y
θ
=
]
[
Y
E
2
]
[
σ
=
Y
V
2
2
()
2
2
1
(, )
2
y
fy
e
θσ
πσ
−
−
=
0
;
;
>
∞
<
<
∞
−
∞
<
<
∞
−
y
4.
Student
t
)
,
(
~
t
Y
=
]
[
Y
E
2
[]
(2
)
VY
ασ
−
for
2
>
=
1
2
2
2
1(
)
(,,)
1
1
,
22
y
B
θσα
σα
+
−
⎛⎞
−
=+
⎜⎟
⎝⎠
5.
Gamma
)
,
(
~
Ga
Y
=
]
[
Y
E
2
]
[
=
Y
V
y
e
y
y
f
−
−
Γ
=
1
)
(
)
,

(
0
;
0
;
0
>
>
>
y
6.
Inverse Gamma
)
,
(
~
IG
Y
1
]
[
−
=
Y
E
)
2
(
)
1
(
]
[
2
2
−
−
=
αα
Y
V
1
1
(,)
y
f
ye
y
+
−
=
Γ
0
;
0
;
0
>
>
>
y
Scale Mixtures Distributions
1.
Scale mixtures of normal form for the Student
t
distribution
)
,
(
~
,

μ
t
X
i
Î
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
i
i
i
N
X
λ
μλ
σμ
2
2
,
~
,
,

and
⎟
⎠
⎞
⎜
⎝
⎛
2
,
2
~
Ga
i
2.
Scale mixtures of uniform form the normal distribution
)
,
(
~
,

2
2
σμσ
N
X
i
Î
( )
i
i
i
i
u
u
U
u
X
+
−
,
~
,
,

2
and
⎟
⎠
⎞
⎜
⎝
⎛
2
,
2
~
Ga
u
i
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