Applied Statistics and Probability for Engineers

184 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

Then, Y 2 X 1  2 X 2 is a normal random variable that represents the perimeter of the

part. We obtain, E(Y)14 centimeters and the variance of Yis

Now,

EXAMPLE 5-38 Soft-drink cans are filled by an automated filling machine. The mean fill volume is 12.1 fluid

ounces, and the standard deviation is 0.1 fluid ounce. Assume that the fill volumes of the cans

are independent, normal random variables. What is the probability that the average volume of

10 cans selected from this process is less than 12 fluid ounces?

Let X 1 , X 2 ,, X 10 denote the fill volumes of the 10 cans. The average fill volume

(denoted as ) is a normal random variable with

Consequently,

EXERCISES FOR SECTION 5-7

P 1 Z3.16 2 0.00079

P 1 X 122 P c

XX

X



12 12.1

1 0.001

d

E 1 X 2 12.1 and V 1 X 2 

0.1^2

10

0.001

X

p

P 1 Z1.12 2 0.13

P 1 Y14.5 2 P 31 YY 2 Y 1 14.5 142 1 0.0416 4

V 1 Y 2  4

 0.1^2  4

 0.2^2 0.0416

5-87. If Xand Yare independent, normal random variables

with E(X)0, V(X)4, E(Y)10, and V(Y)9.

Determine the following:

(a) (b)

(c) (d)

5-88. Suppose that the random variable Xrepresents the

length of a punched part in centimeters. Let Ybe the length

of the part in millimeters. If E(X)5 and V(X)0.25, what

are the mean and variance of Y?

5-89. A plastic casing for a magnetic disk is composed of

two halves. The thickness of each half is normally distributed

with a mean of 2 millimeters and a standard deviation of

0.1 millimeter and the halves are independent.

(a) Determine the mean and standard deviation of the total

thickness of the two halves.

(b) What is the probability that the total thickness exceeds

4.3 millimeters?

5-90. In the manufacture of electroluminescent lamps, sev-

eral different layers of ink are deposited onto a plastic sub-

strate. The thickness of these layers is critical if specifications

regarding the final color and intensity of light of the lamp are

P 12 X 3 Y 302 P 12 X 3 Y 402

E 12 X 3 Y 2 V 12 X 3 Y 2

to be met. Let Xand Ydenote the thickness of two different

layers of ink. It is known that Xis normally distributed with a

mean of 0.1 millimeter and a standard deviation of 0.00031

millimeter and Yis also normally distributed with a mean of

0.23 millimeter and a standard deviation of 0.00017 millime-

ter. Assume that these variables are independent.

(a) If a particular lamp is made up of these two inks only,

what is the probability that the total ink thickness is less

than 0.2337 millimeter?

(b) A lamp with a total ink thickness exceeding 0.2405 mil-

limeters lacks the uniformity of color demanded by the

customer. Find the probability that a randomly selected

lamp fails to meet customer specifications.

5-91. The width of a casing for a door is normally distrib-

uted with a mean of 24 inches and a standard deviation of

1 8 inch. The width of a door is normally distributed with a

mean of 23 and 78 inches and a standard deviation of 1 16

inch. Assume independence.

(a) Determine the mean and standard deviation of the differ-

ence between the width of the casing and the width of the

door.

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