Applied Statistics and Probability for Engineers

5-2

We will also state this result as follows.

A very important application of Equation S5-2 is in finding the distribution of a random vari-

able Y 1 that is a function of two other random variables X 1 and X 2. That is, let Y 1  h 1 (X 1 , X 2 )

where X 1 and X 2 are discrete random variables with joint distribution We want to

find the probability distribution of Y 1 , say, To do this, we define a second function Y 2 

h 2 (X 1 ,X 2 ) so that the one-to-one correspondence between the points (x 1 , x 2 ) and (y 1 , y 2 ) is main-

tained, and we use the result in Equation S5-2 to find the joint probability distribution of Y 1 and

Y 2. Then the distribution of Y 1 alone is found by summing over the y 2 values in this joint distribu-

tion. That is, is just the marginal probability distributionof Y 1 , or

EXAMPLE S5-2 Consider the case where X 1 and X 2 are independent Poisson random variables with parameters

 1 and  2 , respectively. We will find the distribution of the random variable Y 1  X 1  X 2.

The joint distribution of X 1 and X 2 is

because X 1 and X 2 are independent. Now to use Equation S5-2 we need to define a second func-

tionY 2  h 2 (X 1 , X 2 ). Let this function be Y 2  X 2. Now the solutions for x 1 and x 2 are x 1 y 1  y 2

and x 2  y 2. Thus, from Equation S5-2 the joint probability distribution of Y 1 and Y 2 is

Because x 1 0, the transformation x 1  y 1 y 2 requires that x 2  y 2 must always be less than

or equal to y 1. Thus, the values of y 2 are 0, 1,... , y 1 , and the marginal probability distribution

of Y 1 is obtained as follows:

fY 11 y 12  a

y 1

y 2  0

fY 1 Y 21 y 1 , y 22  a

y 1

y 2  0

e^1 ^1 ^22  11 y^1 y^22  2 y^2

1 y 1 y 22! y 2!

fY 1 Y 21 y 1 , y 22 

e^1 ^1 ^22  11 y^1 y^22  2 y^2

1 y 1 y 22! y 2!

, y 1 0, 1, 2, p , y 2 0, 1, p , y 1



e^1 ^1 ^22 x 11 x 22

x 1! x 2!

, x 1 0, 1, p , x 2 0, 1, p



e^1 x 11

x 1!

e^2 x 22

x 2!

fX 1 X 21 x 1 , x 22 fX 11 x 12 fX 21 x 22

fY 11 y 12 a

y 2

fY 1 Y 21 y 1 , y 22

fY 11 y 12

fY 11 y 12.

fX 1 X 21 x 1 , x 22.

Suppose that X 1 and X 2 are discreterandom variables with joint probability distribu-

tion and let Y 1 h 1 (X 1 , X 2 ) and Y 2 h 2 (X 1 , X 2 ) define one-to-one trans-

formations between the points (x 1 , x 2 ) and (y 1 , y 2 ) so that the equations y 1 h 1 (x 1 , x 2 )

and y 2 h 2 (x 1 , x 2 ) can be solved uniquely for x 1 and x 2 in terms of y 1 and y 2. Let this

solution be x 1 u 1 (y 1 , y 2 ) and x 2 u 2 (y 1 , y 2 ). Then the joint probability distribution

of Y 1 and Y 2 is

fY 1 Y 21 y 1 , y 22 fX 1 X 23 u 11 y 1 , y 22 , u 21 y 1 , y 224 (S5-2)

fX 1 X 21 x 1 , x 22 ,

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