Problem (Page 157 # 1, Section 3.7)
Suppose that three random variables X
1
,
X
2
, and X
3
have a continuous joint distribution with the
following joint p.d.f.:
f
(
x
1
,
x
2
,
x
3) =
(
c
(
x
1 + 2
x
2 + 3
x
3)
for
0
≤
x
i
≤
1
i
= 1
,
2
,
3
,
0
otherwise.
Determine
(a)
the value of the constant c;
(b)
the marginal joint p.d.f. of X
1
and X
3
; and
(c)
Pr
`
X
3
<
1
2
˛
˛
X
1 =
1
4
,
X
2 =
3
4
´
.
Solution:
(a)
We have
Z
1
0
Z
1
0
Z
1
0
f
(
x
1
,
x
2
,
x
3)
dx
1
dx
2
dx
3 = 3
c
.
Since the value of this integral must be equal to 1, it follows that
c
= 1
/
3.
(b)
For 0
≤
x
1
≤
1 and 0
≤
x
3
≤
1,
f
13(
x
1
,
x
3) =
Z
1
0
f
(
x
1
,
x
2
,
x
3)
dx
2 =
1
3
(
x
1 + 1 + 3
x
3)
.
(c)
The conditional p.d.f. of
x
3 given that
x
1 =
1
4
and
x
2 =
3
4
is, for 0
≤
x
3
≤
1,
g
(
x
3

x
1 =
1
4
,
x
2 =
3
4
«
=
f
`
1
4
,
3
4
,
x
3
´
f
12
`
1
4
,
3
4
´
=
7
13
+
12
13
x
3
.
Therefore,
Pr
„
X
3
<
1
2

X
1 =
1
4
,
X
2 =
3
4
«
=
∈
1
2
0
„
7
13
+
12
13
x
3
«
dx
3 =
5
13
.
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