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
) =
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(b) To find the
P
[
X
≤
Y
]
we look to integrate over the area indicated by the graph
P
[
X
≤
Y
] =
1
/
2
0
1

x
x
dydx
=
1
/
2
0
(
2

4
x
)
dx
=
1
/
2
Y
X
X=Y
1
1
X
Y
£
(c) The probability
P
[
X
+
Y
≤
1
/
2
]
can be seen in the figure at right. Here we can set up the fol
lowing integrals
P
[
X
+
Y
≤
1
/
2
]
=
1
/
2
0
1
/
2

x
0
2
dydx
=
1
/
2
0
(
1

2
x
)
dx
=
1
/
2

1
/
4
=
1
/
4
Y
X
Y + X = 1
Y + X = ½
1
1
Problem 5.3.1
(a) The joint PDF (and the corresponding region of nonzero probability) are
f
X
,
Y
(
x
,
y
) =
1
/
2

1
≤
x
≤
y
≤
1
0
otherwise
Y
X
1
1
(b)
P
[
X
>
0
] =
1
0
1
x
1
2
dydx
=
1
0
1

x
2
dx
=
1
/
4
This result can be deduced by geometry. The shaded triangle of the
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 MarshallSylvan
 Variance, Probability theory, probability density function, dx

Click to edit the document details