Page 153 - Fundamentals of Probability and Statistics for Engineers

136 Fundamentals of Probability and Statistics for Engineers

Upon evaluating Y (t), the moments of Y are given by [Equation (4.52)]:

n n†
n
EfY gˆ j 0†; n ˆ 1; 2; ... : 5:33†
Y

Example 5.9. Problem: a random variable X is discrete and its pmf is given in

Example 5.1. Determine the mean and variance of Y where Y ˆ 2X ‡ 1.
Answer: using the first of Equations (5.31), we obtain
X
EfYgˆ Ef2X ‡ 1gˆ 2x i ‡ 1†p x i †
X
i

1 1 1 1
ˆ
1† ‡ 1† ‡ 3† ‡ 5† 5:34†
2 4 8 8
3
ˆ ;
4
2
2
2
EfY gˆ Ef 2X ‡ 1† gˆ X 2x i ‡ 1† p x i †
X
i

1 1 1 1 5:35†
ˆ 1† ‡ 1† ‡ 9† ‡ 25†
2 4 8 8
ˆ 5;
and
3
2 71
2
2
2
ˆ EfY g
E fYgˆ 5
ˆ : 5:36†
Y
4 16
Following the second approach, let us use the method of characteristic func-
tions described by Equations (5.32) and (5.33). The characteristic function of Y is
X jt 2x i ‡1†
Y t†ˆ e p x i †
X
i

1 jt 1 3jt 1 5jt 1

jt
ˆ e ‡ e ‡ e ‡ e
2 4 8 8
1
jt jt 3jt 5jt
ˆ 4e ‡ 2e ‡ e ‡ e †;
8
and we have
3
j

1 1†
EfYgˆ j 0†ˆ j
1
4 ‡ 2 ‡ 3 ‡ 5†ˆ ;
Y
8 4
1
2†
2
EfY gˆ
0†ˆ 4 ‡ 2 ‡ 9 ‡ 25†ˆ 5:
Y
8
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