p. 6-21
Proof
. Let
i
=1, …,
n
, then
A
i
(
y
)=
{
x
:
g
i
(
x
)
≤
y
}
,
F
Y
(
y
1
,...,y
n
P
(
Y
1
≤
y
1
,...,Y
n
≤
y
n
)
=
P
(
X
1
∈
A
1
(
y
1
)
,...,X
n
∈
A
n
(
y
n
))
=
P
(
X
1
∈
A
1
(
y
1
))
×···×
P
(
X
n
∈
A
n
(
y
n
))
=
P
(
Y
1
≤
y
1
)
P
(
Y
n
≤
y
n
)
=
F
Y
1
(
y
1
)
F
Y
n
(
y
n
)
.
• Theorem.
X
=(
X
1
, …,
X
n
) are independent if and only if there
exist univariate functions
g
i
(
x
),
i
=1, …,
n
, such that
(a) when
X
1
, …,
X
n
are discrete r.v.’s with joint pmf
p
X
,
p
X
(
x
1
, …,
x
n
)
∝
g
1
(
x
1
)
×
×
g
n
(
x
n
),
∞
<
x
i
<
∞
,
i
=1,…,
n
.
(b) when
X
1
, …,
X
n
are continuous r.v.’s with joint pdf
f
X
,
f
X
(
x
1
, …,
x
n
)
∝
g
1
(
x
1
)
×
×
g
n
(
x
n
),
∞
<
x
i
<
∞
,
i
=1,…,
n
.
p. 6-22
Example.
If the joint pdf of (
X
,
Y
) is
and
f
(
x
,
y
)=0, otherwise, i.e.,
then
X
and
Y
are independent. Note that the region in which
the joint pdf is nonzero can be expressed in the form
{(
x
,
y
):
x
∈
A
,
y
∈
B
}.
f
(
x, y
)
∝
e
−
2
x
e
−
3
y
,
0
<x<
∞
,
0
<y<
∞
,
X
Y