Applied Statistics and Probability for Engineers

EXAMPLE 5-7 For the random variables in Example 5-1, the conditional mean of Ygiven X2 is obtained

from the conditional distribution in Fig. 5-3:

The conditional mean is interpreted as the expected number of acceptable bits given that two

of the four bits transmitted are suspect. The conditional variance of Ygiven X 2 is

5-1.4 Independence

In some random experiments, knowledge of the values of Xdoes not change any of the prob-

abilities associated with the values for Y.

EXAMPLE 5-8 In a plastic molding operation, each part is classified as to whether it conforms to color and

length specifications. Define the random variable Xand Yas

Assume the joint probability distribution of Xand Yis defined by fXY(x, y) in Fig. 5-4(a).

The marginal probability distributions of Xand Yare also shown in Fig. 5-4(a). Note that

fXY(x,y)  fX(x) fY(y). The conditional probability mass function is shown in Fig.

5-4(b). Notice that for any x, fYx(y)  fY(y). That is, knowledge of whether or not the part meets

color specifications does not change the probability that it meets length specifications.

By analogy with independent events, we define two random variables to be independent

whenever fXY(x,y)  fX(x) fY(y) for all xand y. Notice that independence implies that

fXY(x,y)  fX(x) fY(y) for all xand y. If we find one pair of xand yin which the equality fails,

Xand Yare not independent. If two random variables are independent, then

With similar calculations, the following equivalent statements can be shown.

fYƒx 1 y 2 

fXY 1 x, y 2

fX 1 x 2



fX 1 x 2 fY 1 y 2

fX 1 x 2

fY 1 y 2

fYƒx 1 y 2

Ye

1 if the part conforms to length specifications

0 otherwise

Xe

1 if the part conforms to color specifications

0 otherwise

V 1 Y 022  10 Yƒ 2221 0.040 2  11 Yƒ 2221 0.320 2  12 Yƒ 2221 0.640 2 0.32

E 1 Y^022 Yƒ 2  01 0.040 2  11 0.320 2  21 0.640 2 1.6

148 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

x

y

0.0198

0.9702

0 1

0

1

0.0002

0.0098

(a)

0.02

0.98

0.01 0.99

x

y

0.02

0.98

0 1

0

1

0.02

0.98

(b)

fX(x) =

Figure 5-4 (a) Joint fY(y) =

and marginal probabil-

ity distributions ofX

andYin Example 5-8.

(b) Conditional proba-

bility distribution ofY

givenXxin

Example 5-8.

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