Fundamentals of Probability and Statistics for Engineers

5.1.1.1 Discrete Random Variables

Let us fir st 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 g(x 1 ), y 2
g(x 2 ),.... Let the pmf of X be given by

The pmf of y is simply determined as

Ex ample 5. 1. Problem: the pmf of a random variable X is given as

Determine the pmf of Y if Y is related to X by Y 2 X 1.
Answer: the corresponding values of Y are: g( 1) 2( 1) 1 1;
g(0) 1; g(1) 3; and g(2) 5. Hence, the pmf of Y is given by

Ex ample 5. 2. Problem: for the same X as given in Example 5.1, determine the
pmf of Y if Y 2 X^2 1.
Answer: in this case, the corresponding values of Y are: g( 1) 2( 1)^2
1 3; g(0) 1; g(1) 3; and g(2) 9, resulting in

Functions of Random Variables 121

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