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

Definition
Given random variables X , X , p , X and constants c , c , p , c ,
p
2
p
1
1
2
Y c 1 X 1 c 2 X 2 p c p X p (5-36)
is a linear combination of X 1 , X 2 , p , X p .

Now, E(Y) can be found from the joint probability distribution of X 1 , X 2 , p , X p as follows.
Assume X 1 , X 2 , p , X p are continuous random variables. An analogous calculation can be used
for discrete random variables.

p
E1Y2 p 1c x c x c x 2 f X 1 X 2 p X p 1x , x , p , x 2 dx dx p dx p
2
1
p
2 2
1
p p
2
1 1

1
c p x f 1x , x , p , x 2 dx dx p dx p
2
p
1
1
2
1 X 1 X 2 p X p

2
c p x f 1x , x , p , x 2 dx dx p dx , p ,
2
p
2
1
1
p
2 X 1 X 2 p X p

c p

x f
p
p

2
1
2
1
p X 1 X 2 p X p 1x , x , p , x 2 dx dx p dx p

By using Equation 5-24 for each of the terms in this expression, we obtain the following.
Mean of a
Linear If Y c X c X p c X ,
1 1
2 2
p
Combination p
E1Y2 c E 1X 2 c E 1X 2 p c E 1X 2 (5-37)
1
1
2
p
p
2
Furthermore, it is left as an exercise to show the following.