Applied Statistics and Probability for Engineers

180 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

An important use of the bivariate normal distribution is to calculate probabilities involving

two correlated normal random variables.

EXAMPLE 5-34 Suppose that the Xand Ydimensions of an injection-molded part have a bivariate normal

distribution with X0.04, Y0.08. X3.00. Y7.70, and 0.8. Then, the prob-

ability that a part satisfies both specifications is

This probability can be obtained by integrating fXY(x,y;X, Y,XY, ) over the region

2.95x3.05 and 7.60y7.80, as shown in Fig. 5-7. Unfortunately, there is often no

closed-form solution to probabilities involving bivariate normal distributions. In this case, the

integration must be done numerically.

EXERCISES FOR SECTION 5-6

P 1 2.95X3.05, 7.60Y7.80 2

5-79. Let Xand Yrepresent concentration and viscosity of a

chemical product. Suppose Xand Yhave a bivariate normal

distribution with X4, Y1, X2, and Y1. Draw

a rough contour plot of the joint probability density function

for each of the following values for :

(a) 0 (b)0.8

(c)0.8

5-80. Let Xand Yrepresent two dimensions of an injec-

tion molded part. Suppose Xand Yhave a bivariate normal

distribution with X0.04, Y0.08, X3.00,

Y7.70, and Y0. Determine P(2.95X3.05,

7.60Y7.80).

5-81. In the manufacture of electroluminescent lamps,

several different layers of ink are deposited onto a plastic

substrate. The thickness of these layers is critical if specifi-

cations regarding the final color and intensity of light of

the lamp are to be met. Let Xand Ydenote the thickness

of two different layers of ink. It is known that Xis nor-

mally 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 millimeter. The value of for

these variables is equal to zero. Specifications call for a

lamp to have a thickness of the ink corresponding to Xin

the range of 0.099535 to 0.100465 millimeters and Yin

the range of 0.22966 to 0.23034 millimeters. What is the

probability that a randomly selected lamp will conform to

specifications?

5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES

A random variable is sometimes defined as a function of one or more random variables.

The CD material presents methods to determine the distributions of general functions of

random variables. Furthermore, moment-generating functions are introduced on the CD

5-82. Suppose that Xand Yhave a bivariate normal distri-

bution with joint probability density function fXY(x,y;X, Y,

X, Y, ).

(a) Show that the conditional distribution of Y, given that

Xxis normal.

(b) Determine.

(c) Determine.

5-83. If Xand Yhave a bivariate normal distribution with

0, show that Xand Yare independent.

5-84. Show that the probability density function fXY(x,y;

X, Y, X, Y, ) of a bivariate normal distribution integrates

to one. [Hint:Complete the square in the exponent and use the

fact that the integral of a normal probability density function

for a single variable is 1.]

5-85. If Xand Yhave a bivariate normal distribution with

joint probability density fXY(x,y;X, Y, X,Y, ), show

that the marginal probability distribution of Xis normal

with mean Xand standard deviation X. [Hint:Complete

the square in the exponent and use the fact that the integral

of a normal probability density function for a single variable

is 1.]

5-86. If Xand Yhave a bivariate normal distribution with

joint probability density fXY(x,y;X, Y, X, Y, ), show that

the correlation between Xand Yis . [Hint:Complete the

square in the exponent].

V 1 Y 0 Xx 2

E 1 Y 0 Xx 2

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