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5-8 FUNCTIONS OF RANDOM VARIABLES (CD ONLY)

In many situations in statistics, it is necessary to derive the probability distribution of a func-
tion of one or more random variables. In this section, we present some results that are helpful
in solving this problem.
Suppose that X is a discrete random variable with probability distribution f (x). Let Y h(X)
X
be a function of X that defines a one-to-one transformation between the values of X and Y, and we
wish to find the probability distribution of Y. By a one-to-one transformation, we mean that each
value x is related to one and only one value of y h(x) and that each value of y is related to one
and only one value of x, say, x u(y), where u(y) is found by solving y h(x) for x in terms of y.
Now, the random variable Y takes on the value y when X takes on the value u( y).
Therefore, the probability distribution of Y is

f 1 y2 P1Y y2 P3X u1 y24 f 3u1 y24
Y X
We may state this result as follows.

Suppose that X is a discrete random variable with probability distribution f X (x). Let
Y h(X) define a one-to-one transformation between the values of X and Y so that
the equation y h(x) can be solved uniquely for x in terms of y. Let this solution be
x u(y). Then the probability distribution of the random variable Y is

1 y2 f 3u1 y24
X
f Y (S5-1)

EXAMPLE S5-1 Let X be a geometric random variable with probability distribution

f X 1x2 p 11 p2 x 1 , x 1, 2, p

2
Find the probability distribution of Y X .
2
Since X 0, the transformation is one to one; that is, y x and x 1y. Therefore,
Equation S5-1 indicates that the distribution of the random variable Y is
1 y2 f 11y2 p 11 p2 1y 1 , y 1, 4, 9, 16, p
f Y
Now suppose that we have two discrete random variables X 1 and X 2 with joint probability
1x 1 , x 2 2 and we wish to find the joint probability distribution f 1y 1 , y 2 2 of
Y 1 Y 2
distribution f X 1 X 2
two new random variables Y 1 h 1 (X 1 , X 2 ) and Y 2 h 2 (X 1 , X 2 ). We assume that the functions
h 1 and h 2 define a one-to-one transformation between (x 1 , x 2 ) and (y 1 , y 2 ). Solving the equa-
tions y 1 h 1 (x 1 , x 2 ) and y 2 h 2 (x 1 , x 2 ) simultaneously, we obtain the unique solution
x 1 u 1 (y 1 , y 2 ) and x 2 u 2 (y 1 , y 2 ). Therefore, the random variables Y 1 and Y 2 take on the
values y 1 and y 2 when X 1 takes on the value u 1 (y 1 , y 2 ) and X 2 takes the value u 2 (y 1 , y 2 ). The
joint probability distribution of Y 1 and Y 2 is

f 1 y , y 2 P1Y y , Y y 2
Y 1 Y 2 1 2 1 1 2 2
P3X u 1y , y 2, X u 1 y , y 24
2
1
2
2
1
1
2
1
3u 1 y , y 2, u 1y , y 24
f X 1 X 2 1 1 2 2 1 2
5-1