Probability and Stochastic Processes:
A Friendly Introduction for Electrical and Computer Engineers
Edition 2
Roy D. Yates and David J. Goodman
Problem Solutions
: Yates and Goodman,
3.5.3 3.5.4 3.5.7 3.5.8 3.5.10 3.7.2 3.7.5 3.7.7
3.7.11 and 3.7.16
Problem 3.5.3 Solution
X
is a Gaussian random variable with zero mean but unknown variance.
We do know,
however, that
P
[

X
 ≤
10] = 0
.
1
(1)
We can find the variance Var[
X
] by expanding the above probability in terms of the Φ(
·
)
function.
P
[

10
≤
X
≤
10] =
F
X
(10)

F
X
(

10) = 2Φ
10
σ
X

1
(2)
This implies Φ(10
/σ
X
) = 0
.
55. Using Table 3.1 for the Gaussian CDF, we find that 10
/σ
X
=
0
.
15 or
σ
X
= 66
.
6.
Problem 3.5.4 Solution
Repeating Definition 3.11,
Q
(
z
) =
1
√
2
π
Z
∞
z
e

u
2
/
2
du
(1)
Making the substitution
x
=
u/
√
2, we have
Q
(
z
) =
1
√
π
Z
∞
z/
√
2
e

x
2
dx
=
1
2
erfc
z
√
2
(2)
Problem 3.5.7 Solution
We are given that there are 100
,
000
,
000 men in the United States and 23
,
000 of them are
at least 7 feet tall, and the heights of U.S men are independent Gaussian random variables
with mean 5
0
10
00
.
(a) Let
H
denote the height in inches of a U.S male.
To find
σ
X
, we look at the fact
that the probability that
P
[
H
≥
84] is the number of men who are at least 7 feet
tall divided by the total number of men (the frequency interpretation of probability).
Since we measure
H
in inches, we have
P
[
H
≥
84] =
23
,
000
100
,
000
,
000
= Φ
70

84
σ
X
= 0
.
00023
(1)
Since Φ(

x
) = 1

Φ(
x
) =
Q
(
x
),
Q
(14
/σ
X
) = 2
.
3
·
10

4
(2)
From Table 3.2, this implies 14
/σ
X
= 3
.
5 or
σ
X
= 4.
1
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(b) The probability that a randomly chosen man is at least 8 feet tall is
P
[
H
≥
96] =
Q
96

70
4
=
Q
(6
.
5)
(3)
Unfortunately, Table 3.2 doesn’t include
Q
(6
.
5), although it should be apparent that
the probability is very small. In fact,
Q
(6
.
5) = 4
.
0
×
10

11
.
(c) First we need to find the probability that a man is at least 7’6”.
P
[
H
≥
90] =
Q
90

70
4
=
Q
(5)
≈
3
·
10

7
=
β
(4)
Although Table 3.2 stops at
Q
(4
.
99), if you’re curious, the exact value is
Q
(5) =
2
.
87
·
10

7
.
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 Spring '08
 Duman
 Normal Distribution, Variance, Probability theory, Exponential distribution

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