Applied Statistics and Probability for Engineers

5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 181

and used to determine the distribution of a sum of random variables. In this section, results

for linear functions are highlighted because of their importance in the remainder of the

book. References are made to the CD material as needed. For example, if the random vari-

ables X 1 and X 2 denote the length and width, respectively, of a manufactured part, Y 2 X 1

 2 X 2 is a random variable that represents the perimeter of the part. As another example,

recall that the negative binomial random variable was represented as the sum of several

geometric random variables.

In this section, we develop results for random variables that are linear combinations of

random variables.

Given random variables X 1 , X 2 ,, Xpand constants c 1 , c 2 ,, cp,

(5-36)

is a linear combinationof X 1 , X 2 ,p, Xp.

Yc 1 X 1 c 2 X 2 pcpXp

p p

Definition

Now, E(Y) can be found from the joint probability distribution of X 1 , X 2 ,, Xpas follows.

Assume X 1 , X 2 ,, Xpare continuous random variables. An analogous calculation can be used

for discrete random variables.

By using Equation 5-24 for each of the terms in this expression, we obtain the following.

cp (^) 

(^) 


p 



xp fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxp

c (^2) 

(^) 


p 



x 2 fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxp,p,

c (^1) 

(^) 


p 



x 1 fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxp

E 1 Y 2  



(^) 


p 



1 c 1 x 1 c 2 x 2 pcpxp 2 fX 1 X 2 p Xp 1 x 1 , x 2 ,p, xp 2 dx 1 dx 2 pdxp

p

p

If

E 1 Y 2 c 1 E 1 X 12 c 2 E 1 X 22 pcpE 1 Xp 2 (5-37)

Yc 1 X 1 c 2 X 2 pcp Xp,

Mean of a

Linear

Combination

Furthermore, it is left as an exercise to show the following.

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