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5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 183

Mean and
Variance of an If X 1X X p X 2 p with E(X ) for i 1, 2, p , p
i
1
p
2
Average
E1X 2 (5-40a)
2
2
i
if X 1 , X , p , X are also independent with V(X ) for i 1, 2, p , p,
p
2
V1X 2 (5-40b)
p

The conclusion for V1X 2 is obtained as follows. Using Equation 5-39, with c 1 p and
i
2
V(X i ) , yields
2
2 2
2 2
V1X 2 11 p2 p 11 p2 p
w
p terms
Another useful result concerning linear combinations of random variables is a reproduc-
tive property that holds for independent, normal random variables.

Reproductive
Property of the
If X , X , p , X are independent, normal random variables with E(X ) and
i
i
2
1
p
Normal 2
V1X 2 i , for i 1, 2, p , p,
i
Distribution
Y c X c X p c X p
2
1 1
2
p
is a normal random variable with
E1Y 2 c c p c
2 2
p p
1 1
and
2 2
2 2
2 2
V1Y 2 c 1 1 c 2 2 p c p p (5-41)
The mean and variance of Y follow from Equations 5-37 and 5-39. The fact that Y has a nor-
mal distribution can be obtained from moment-generating functions discussed in Section 5-9
in the CD material.
EXAMPLE 5-37 Let the random variables X 1 and X 2 denote the length and width, respectively, of a manufac-
tured part. Assume that X 1 is normal with E(X 1 ) 2 centimeters and standard deviation
0.1 centimeter and that X 2 is normal with E(X 2 ) 5 centimeters and standard deviation 0.2
centimeter. Also, assume that X 1 and X 2 are independent. Determine the probability that the
perimeter exceeds 14.5 centimeters.