5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 183
The conclusion for is obtained as follows. Using Equation 5-39, with and
V(Xi)^2 , yields
Another useful result concerning linear combinations of random variables is a reproduc-
tive propertythat holds for independent, normal random variables.
p terms
V 1 X 2 11 p 22 ^2 p 11 p 22 ^2 ^2 p
V 1 X 2 ci 1 p
If with E(Xi)for i1, 2, , p
(5-40a)
if X 1 , X 2 ,, Xpare also independent with V(Xi)^2 for i1, 2, , p,
V 1 X 2 (5-40b)
^2
p
p p
E 1 X 2
X 1 X 1 X 2 pXp 2 p p
Mean and
Variance of an
Average
w
If X 1 , X 2 ,, Xpare independent, normal random variables with E(Xi)iand
, for i1, 2, , p,
is a normal random variable with
and
V 1 Y 2 c^21 ^21 c^22 22 pc^2 pp^2 (5-41)
E 1 Y 2 c 1 1 c 2 2 pcpp
Yc 1 X 1 c 2 X 2 pcpXp
V 1 Xi 2 ^2 i p
p
Reproductive
Property of the
Normal
Distribution
The mean and variance of Yfollow from Equations 5-37 and 5-39. The fact that Yhas 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.
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