Probability and Stochastic Processes:
A Friendly Introduction for Electrical and Computer Engineers
Roy D. Yates and David J. Goodman
Problem Solutions
: Yates and Goodman,5.1.2 5.2.1 5.3.1 5.3.2 5.5.1 5.5.2 5.6.1 5.6.2 5.7.1 and
5.8.1
Problem 5.1.2
(a) Because the probability that any random variable is less than

∞
is zero, we have
F
X
,
Y
(
x
,

∞
) =
P
[
X
≤
x
,
Y
≤ 
∞
]
≤
P
[
Y
≤ 
∞
] =
0
(b) The probability that any random variable is less than infinity is always one.
F
X
,
Y
(
x
,
∞
) =
P
[
X
≤
x
,
Y
≤
∞
] =
P
[
X
≤
x
] =
F
X
(
x
)
(c) Although
P
[
Y
≤
∞
] =
1,
P
[
X
≤ 
∞
] =
0. Therefore the following is true.
F
X
,
Y
(

∞
,
∞
) =
P
[
X
≤ 
∞
,
Y
≤
∞
]
≤
P
[
X
≤ 
∞
] =
0
(d) Part (d) follows the same logic as that of part (a).
F
X
,
Y
(

∞
,
y
) =
P
[
X
≤ 
∞
,
Y
≤
y
]
≤
P
[
X
≤ 
∞
] =
0
(e) Analogous to Part (b), we find that
F
X
,
Y
(
∞
,
y
) =
P
[
X
≤
∞
,
Y
≤
y
] =
P
[
Y
≤
y
] =
F
Y
(
y
)
Problem 5.2.1
(a) The joint PDF of
X
and
Y
is
f
X
,
Y
(
x
,
y
) =
½
c
x
+
y
≤
1
,
x
,
y
≥
0
0
otherwise
Y
X
Y + X = 1
1
1
To find the constant
c
we integrate over the region shown. This gives
1
0
1

x
0
cdydx
=
cx

cx
2
¯
¯
¯
1
0
=
c
2
=
1
Therefore
c
=
2.
1