Fundamentals of Probability and Statistics for Engineers

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

Ex ample 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 2 X 1.
Answer: using the first of Equations (5.31), we obtain

and

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

and we have

136 Fundamentals of Probability and Statistics for Engineers

EfYngˆj^ n…Yn†… 0 †;nˆ 1 ; 2 ;...:… 5 : 33 †

ˆ‡

EfYgˆEf 2 X‡ 1 gˆ

X

i

… 2 xi‡ 1 †pX…xi†

ˆ…

1 †

1

2



‡… 1 †

1

4



‡… 3 †

1

8



‡… 5 †

1

8



ˆ

3

4

;

… 5 : 34 †

EfY^2 gˆEf… 2 X‡ 1 †^2 gˆ

X

i

… 2 xi‡ 1 †^2 pX…xi†

ˆ… 1 †

1

2



‡… 1 †

1

4



‡… 9 †

1

8



‡… 25 †

1

8



ˆ 5 ;

… 5 : 35 †

^2 YˆEfY^2 g

E^2 fYgˆ 5

3

4

 2

ˆ

71

16

: … 5 : 36 †

Y…t†ˆ

X

i

ejt…^2 xi‡^1 †pX…xi†

ˆe^ jt

1

2



‡ejt

1

4



‡e3jt

1

8



‡e5jt

1

8



ˆ

1

8

…4e^ jt‡2ejt‡e3jt‡e5jt†;

EfYgˆj^1 …Y^1 †… 0 †ˆj^1

j

8



…

4 ‡ 2 ‡ 3 ‡ 5 †ˆ

3

4

;

EfY^2 gˆ

…Y^2 †… 0 †ˆ

1

8

… 4 ‡ 2 ‡ 9 ‡ 25 †ˆ 5 :