•
A marginal cdf (pdf, pmf) is the joint cdf (pdf, pmf) for a subset of
{
X
1
, . . . , X
n
}
; e.g., for
X
=
X
1
X
2
X
3
the marginals are
f
X
1
(
x
1
)
, f
X
2
(
x
2
)
, f
X
3
(
x
3
)
f
X
1
,X
2
(
x
1
, x
2
)
, f
X
1
,X
3
(
x
1
, x
3
)
, f
X
2
,X
3
(
x
2
, x
3
)
•
The marginals can be obtained from the joint in the usual way. For the previous
example,
F
X
1
(
x
1
) =
lim
x
2
,x
3
→∞
F
X
(
x
1
, x
2
, x
3
)
f
X
1
,X
2
(
x
1
, x
2
) =
Z
∞
∞
f
X
1
,X
2
,X
3
(
x
1
, x
2
, x
3
)
dx
3
EE 278: Random Vectors
6 – 3
•
Conditional cdf (pdf, pmf) can also be defined in the usual way. E.g., the
conditional pdf of
X
n
k
+1
= (
X
k
+1
, . . . , X
n
)
given
X
k
= (
X
1
, . . . , X
k
)
is
f
X
n
k
+1

X
k
(
x
n
k
+1

x
k
) =
f
X
(
x
1
, x
2
, . . . , x
n
)
f
X
k
(
x
1
, x
2
, . . . , x
k
)
=
f
X
(
x
)
f
X
k
(
x
k
)
•
Chain Rule
: We can write
f
X
(
x
) =
f
X
1
(
x
1
)
f
X
2

X
1
(
x
2

x
1
)
f
X
3

X
1
,X
2
(
x
3

x
1
, x
2
)
· · ·
f
X
n

X
n

1
(
x
n

x
n

1
)
Proof: By induction. The chain rule holds for
n
= 2
by definition of conditional
pdf. Now suppose it is true for
n

1
. Then
f
X
(
x
) =
f
X
n

1
(
x
n

1
)
f
X
n

X
n

1
(
x
n

x
n

1
)
=
f
X
1
(
x
1
)
f
X
2

X
1
(
x
2

x
1
)
· · ·
f
X
n

1

X
n

2
(
x
n

1

x
n

2
)
f
X
n

X
n

1
(
x
n

x
n

1
)
,
which completes the proof
EE 278: Random Vectors
6 – 4