Applied Statistics and Probability for Engineers

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 Xis a discrete random variable with probability distribution fX(x). Let Y h(X)

be a function of Xthat defines a one-to-one transformation between the values of Xand Y, and we

wish to find the probability distribution of Y. By a one-to-one transformation, we mean that each

value xis related to one and only one value of y h(x) and that each value of yis related to one

and only one value of x, say, x u(y), where u(y) is found by solving y h(x) for xin terms of y.

Now, the random variable Ytakes on the value ywhen Xtakes on the value u(y).

Therefore, the probability distribution of Yis

We may state this result as follows.

fY 1 y 2 P 1 Yy 2 P 3 Xu 1 y 24 fX 3 u 1 y 24

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

Find the probability distribution of Y X^2.

Since X0, the transformation is one to one; that is, y x^2 and Therefore,

Equation S5-1 indicates that the distribution of the random variable Yis

Now suppose that we have two discrete random variables X 1 and X 2 with joint probability

distribution and we wish to find the joint probability distribution of

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

fX 1 X 23 u 11 y 1 , y 22 , u 21 y 1 , y 224

P 3 X 1 u 11 y 1 , y 22 , X 2 u 21 y 1 , y 224

fY 1 Y 21 y 1 , y 22 P 1 Y 1 y 1 , Y 2 y 22

fX 1 X 2 1 x 1 , x 22 fY 1 Y 21 y 1 , y 22

fY 1 y 2 f 11 y 2 p 11 p 21 y^1 , y1, 4, 9, 16,p

x 1 y.

fX 1 x 2 p 11 p 2 x^1 , x1, 2,p

5-1

Suppose that Xis a discreterandom variable with probability distribution fX(x). Let

Yh(X) define a one-to-one transformation between the values of Xand Yso that

the equation yh(x) can be solved uniquely for xin terms of y. Let this solution be

xu(y). Then the probability distribution of the random variable Yis

fY 1 y 2 fX 3 u 1 y (^24) (S5-1)
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