Applied Statistics and Probability for Engineers

68 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Now

Although Xnever assumes the value 0.4, the weighted average of the possible values is 0.4.

To calculate V 1 X 2 ,a table is convenient.

0.4

 01 0.6561 2   11 0.2916 2 21 0.0486 2 31 0.0036 2 41 0.0001 2

E 1 X 2  0 f 102 1 f 112 2 f 122 3 f 132 4 f 142

0 0.4 0.16 0.6561 0.104976

1 0.6 0.36 0.2916 0.104976

2 1.6 2.56 0.0486 0.124416

3 2.6 6.76 0.0036 0.024336

4 3.6 12.96 0.0001 0.001296

x x0.4 1 x0.4 22 f 1 x 2 f 1 x 21 x0.4 22

The alternative formula for variance could also be used to obtain the same result.

EXAMPLE 3-10 Two new product designs are to be compared on the basis of revenue potential. Marketing

feels that the revenue from design A can be predicted quite accurately to be $3 million. The

revenue potential of design B is more difficult to assess. Marketing concludes that there is a

probability of 0.3 that the revenue from design B will be $7 million, but there is a 0.7 proba-

bility that the revenue will be only $2 million. Which design do you prefer?

Let Xdenote the revenue from design A. Because there is no uncertainty in the revenue

from design A, we can model the distribution of the random variable Xas $3 million with

probability 1. Therefore, million.

Let Ydenote the revenue from design B. The expected value of Yin millions of dollars is

Because E(Y) exceeds E(X), we might prefer design B. However, the variability of the result

from design B is larger. That is,

Because the units of the variables in this example are millions of dollars, and because the vari-

ance of a random variable squares the deviations from the mean, the units of are millions

of dollars squared. These units make interpretation difficult.

Because the units of standard deviation are the same as the units of the random variable,

the standard deviation is easier to interpret. In this example, we can summarize our results

as “the average deviation of Yfrom its mean is $2.29 million.’’

2

5.25 millions of dollars squared

2  17 3.5 221 0.3 2  12 3.5 221 0.7 2

E 1 Y 2 $ 71 0.3 2  $ 21 0.7 2 $3.5

E 1 X 2 $ 3

V 1 X 2 ^2  a

5

i 1

f^1 xi^21 xi0.4^2

(^2) 0.36
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