Applied Statistics and Probability for Engineers

182 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

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-

ple. Let X 1 denote any random variable and define X 2 X 1. Clearly, X 1 and X 2 are not inde-

pendent. In fact, XY 1. Now, YX 1 X 2 is 0 with probability 1. Therefore, V(Y)0,

regardless of the variances of X 1 and X 2.

EXAMPLE 5-35 In Chapter 3, we found that if Yis a negative binomial random variable with parameters pand

r, where each Xiis a geometric random variable with parameter

pand they are independent. Therefore, and. From Equation

5-37, and from Equation 5-39,.

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 1 and X 2 denote the length and width, respectively, of a man-

ufactured part. Assume E(X 1 )2 centimeters with standard deviation 0.1 centimeter and

E(X 2 )5 centimeters with standard deviation 0.2 centimeter. Also, assume that the covari-

ance between X 1 and X 2 is0.005. Then, Y 2 X 1  2 X 2 is a random variable that represents

the perimeter of the part. From Equation 5-36,

and from Equation 5-38

Therefore, the standard deviation of Yis

The particular linear combination that represents the average of prandom variables, with

identical means and variances, is used quite often in the subsequent chapters. We highlight the

results for this special case.

0.16^1  2 0.4 centimeters.

0.040.160.040.16 centimeters squared

V 1 Y 2  221 0.1^22  221 0.2^22  2221 0.005 2

E 1 Y 2  2122  2152 14 centimeters

E 1 Y 2 r p V 1 Y 2 r 11 p 2 p^2

E 1 Xi 2  1 p E 1 Xi 2  11 p 2 p^2

YX 1 X 2 pXr,

If X 1 , X 2 ,, Xpare random variables, and then in

general

(5-38)

If X 1 , X 2 ,, Xpare independent,

V 1 Y 2 c^21 V 1 X 12 c^22 V 1 X 22 pc^2 pV 1 Xp 2 (5-39)

p

V 1 Y 2 c^21 V 1 X 12 c^22 V 1 X 22 pc^2 pV 1 Xp 2  (^2) a
ij
(^) a cicj cov 1 Xi, Xj 2
p Yc 1 X 1 c 2 X 2 pcpXp,
Variance of a
Linear
Combination
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