Page 206 - Applied Statistics And Probability For Engineers

c05.qxd 5/13/02 1:50 PM Page 182 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

182 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

Variance of a
Linear p
If X , X , p , X are random variables, and Y c X c X c X , then in
2
2
p
1 1
2
p
p
1
Combination
general
2
2
2
V1Y2 c 1 V1X 2 c 2 V1X 2 p c p V1X 2 2 a a c c cov1X , X 2 (5-38)
1
i j

j
i
p
2
i j
If X , X , p , X are independent,
1
2
p
2
2
2
V1Y 2 c 1 V1X 2 c 2 V1X 2 p c p V1X 2 (5-39)
p
2
1
Note that the result for the variance in Equation 5-39 requires the random variables to be
independent. To see why the independence is important, consider the following simple exam-
denote any random variable and define X X . Clearly, X and X are not inde-
ple. Let X 1 2 1 1 2
pendent. In fact, XY 1. Now, Y X X is 0 with probability 1. Therefore, V(Y) 0,
1
2
regardless of the variances of X and X .
2
1
EXAMPLE 5-35 In Chapter 3, we found that if Y is a negative binomial random variable with parameters p and
r, Y X X p X , where each X is a geometric random variable with parameter
1
r
2
i
p and they are independent. Therefore, E1X 2 1 p and E1X 2 11 p2 p 2 . From Equation
i
i
5-37, E1Y2 r p and from Equation 5-39, V 1Y 2 r11 p2 p 2 .
An approach similar to the one applied in the above example can be used to verify the
formulas for the mean and variance of an Erlang random variable in Chapter 4.
EXAMPLE 5-36 Suppose the random variables X and X denote the length and width, respectively, of a man-
2
1
ufactured part. Assume E(X ) 2 centimeters with standard deviation 0.1 centimeter and
1
E(X ) 5 centimeters with standard deviation 0.2 centimeter. Also, assume that the covari-
2
ance between X and X 2 is 0.005. Then, Y 2X 1 2X 2 is a random variable that represents
1
the perimeter of the part. From Equation 5-36,
E1Y2 2122 2152 14 centimeters
and from Equation 5-38
2
2
2
2
V1Y2 2 10.1 2 2 10.2 2 2 2 21 0.0052
0.04 0.16 0.04 0.16 centimeters squared
Therefore, the standard deviation of Y is 0.16 1 2 0.4 centimeters.
The particular linear combination that represents the average of p random variables, with
identical means and variances, is used quite often in the subsequent chapters. We highlight the
results for this special case.