Applied Statistics and Probability for Engineers

170 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

The concept of independence can be extended to multiple continuous random variables.

Continuous random variables are independentif and only if

fX 1 X 2 p Xp 1 x 1 , x 2 p, xp 2 fX 11 x 12 fX 21 x 22 pfXp 1 xp 2 for all x 1 , x 2 ,p, xp (5-26)

X 1 , X 2 ,p, Xp

Definition

Similar to the result for only two random variables, independence implies that Equation 5-26

holds for all If we find one point for which the equality fails, are

not independent. It is left as an exercise to show that if are independent,

for anyregions in the range of respectively.

EXAMPLE 5-25 In Chapter 3, we showed that a negative binomial random variable with parameters pand r

can be represented as a sum of rgeometric random variables Each geometric

random variable represents the additional trials required to obtain the next success. Because

the trials in a binomial experiment are independent, are independent random

variables.

EXAMPLE 5-26 Suppose and represent the thickness in micrometers of a substrate, an active layer,

and a coating layer of a chemical product. Assume that and are independent

and normally distributed with and

respectively. The specifications for the thickness of the substrate, active layer, and

coating layer are and respectively.

What proportion of chemical products meets all thickness specifications? Which one of the

three thicknesses has the least probability of meeting specifications?

The requested probability is

Because the random variables are independent,

After standardizing, the above equals

where Zis a standard normal random variable. From the table of the standard normal distri-

bution, the above equals

The thickness of the coating layer has the least probability of meeting specifications.

Consequently, a priority should be to reduce variability in this part of the process.

1 0.99862 21 0.98758 21 0.78870 2 0.7778

P 1 3.2Z3.2 2 P 1 2.5Z2.5 2 P 1 1.25Z1.25 2

P 19200 X 1  108002 P 1950 X 2  10502 P 175 X 3  852

P 19200 X 1 10800, 950X 2 1050, 75X 3  852

P 19200 X 1 10800, 950X 2 1050, 75X 3 85.

9200 x 1 10800, 950x 2 1050, 75 x 3 85,

 3 4,

 1 10000,  2 1000,  3 80,  1 250,  2 20,

X 1 , X 2 , X 3

X 1 , X 2 , X 3

X 1 , X 2 ,p, Xr

X 1 , X 2 ,p, Xr.

A 1 , A 2 ,p, Ap X 1 , X 2 ,p, Xp,

P 1 X 1 A 1 , X 2 A 2 ,p, XpAp 2 P 1 X 1 A 12 P 1 X 2 A 22 pP 1 XpAp 2

X 1 , X 2 ,p, Xp

x 1 , x 2 ,p, xp. X 1 , X 2 ,p, Xp

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