Applied Statistics and Probability for Engineers

158 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

At the start of this chapter, the lengths of different dimensions of an injection-molded part

were presented as an example of two random variables. Each length might be modeled by a

normal distribution. However, because the measurements are from the same part, the random

variables are typically not independent. A probability distribution for two normal random vari-

ables that are not independent is important in many applications and it is presented later in this

chapter. If the specifications for Xand Yare 2.95 to 3.05 and 7.60 to 7.80 millimeters, respec-

tively, we might be interested in the probability that a part satisfies both specifications; that is,

Suppose that is shown in Fig. 5-7. The re-

quired probability is the volume of within the specifications. Often a probability such

as this must be determined from a numerical integration.

EXAMPLE 5-15 Let the random variable Xdenote the time until a computer server connects to your machine

(in milliseconds), and let Ydenote the time until the server authorizes you as a valid user (in

milliseconds). Each of these random variables measures the wait from a common starting time

and X Y. Assume that the joint probability density function for Xand Yis

Reasonable assumptions can be used to develop such a distribution, but for now, our focus is

only on the joint probability density function.

The region with nonzero probability is shaded in Fig. 5-8. The property that this joint

probability density function integrates to 1 can be verified by the integral of fXY(x, y) over this

region as follows:

0.003 °

0

e0.003x dx¢0.003 a

1

0.003

b 1

 6 10 ^6 

0

°

e0.002x

0.002

¢ e0.001x dx

 6 10 ^6 

0

°

x

e0.002y dy¢ e0.001x dx

(^) 

(^) 


fXY 1 x, y 2 dy dx

0

°

x

610 ^6 e0.001x0.002y dy¢ dx

fXY 1 x, y 2  6 10 ^6 exp 1 0.001x0.002y 2 for xy

fXY 1 x, y 2

P 1 2.95X3.05, 7.60Y7.80 2. fXY 1 x, y 2

Figure 5-6 Joint probability density function for

random variablesXandY.

fXY(x, y)

x

y

R

Probability that (X, Y) is in the region R is determined

by the volume of fXY(x, y) over the region R.

fXY(x, y)

y

x

3.0

2.95

7.70 3.05

7.80

7.60

Figure 5-7 Joint probability density function for the lengths

of different dimensions of an injection-molded part.

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