Page 138 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 121

5.1.1.1 Discrete Random Variables

Let us first dispose of the case when X is a discrete random variable, since it
requires only simple point-to-point mapping. Suppose that the possible values
taken by X can be enumerated as x 1 , x 2 ,.. .. Equation (5.2) shows that the
corresponding possible values of Y may be enumerated as y 1 ˆ g(x 1 ), y 2 ˆ
g(x 2 ), . . . . Let the pmf of X be given by

p x i †ˆ p i ; i ˆ 1; 2; ... : 5:3†
X
The pmf of y is simply determined as
p y i †ˆ p ‰g x i †Š ˆ p i ; i ˆ 1; 2; ... : 5:4†
Y
Y

Example 5.1. Problem: the pmf of a random variable X is given as
8
1
> ; for x ˆ
1;
2
>
>
>
>
> 1
> ; for x ˆ 0;
<
p x†ˆ 4
X 1
> ; for x ˆ 1;
8
>
>
>
>
1
>
>
: ; for x ˆ 2;
8
Determine the pmf of Y if Y is related to X by Y ˆ 2X ‡ 1.

Answer: the corresponding values of Y are: g( 1) ˆ 2( 1) ‡ 1 ˆ
1;
g(0) ˆ 1; g(1) ˆ 3; and g(2) ˆ 5. Hence, the pmf of Y is given by
1
8
> ; for y ˆ
1;
2
>
>
>
>
> 1
> ; for y ˆ 1;
<
p y†ˆ 4
Y
> 1 ; for y ˆ 3;
>
> 8
>
1
>
>
>
: ; for y ˆ 5.
8

Example 5.2. Problem: for the same X as given in Example 5.1, determine the

pmf of Y if Y ˆ 2X ‡ 1.
2
2
Answer: in this case, the corresponding values of Y are: g( 1) ˆ 2( 1) ‡

1 ˆ 3; g(0) ˆ 1; g(1) ˆ 3; and g(2) ˆ 9, resulting in
1 ; for y ˆ 1;
8
4
>
>
>
<
p y†ˆ 5 ˆ 1 ‡ 1 ; for y ˆ 3;
Y
> 8 2 8
>
1
>
: ; for y ˆ 9:
8
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