Applied Statistics and Probability for Engineers

188 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

5-122. Show that if X 1 , X 2 ,p, Xpare independent,

continuous random variables, P(X 1  A 1 ,X 2  A 2 , p ,

Xp Ap) P(X 1  A 1 )P(X 2  A 2 ) p P(Xp Ap) for any

regions A 1 , A 2 , p , Apin the range of X 1 , X 2 , p , Xp

respectively.

5-123. Show that if X 1 , X 2 ,p, Xpare independent

random variables and Yc 1 X 1  c 2 X 2  cpXp,

You can assume that the random variables are continuous.

5-124. Suppose that the joint probability function of

the continuous random variables Xand Yis constant on

the rectangle 0xa, 0yb. Show that Xand Y

are independent.

5-125. Suppose that the range of the continuous

variables Xand Yis 0xaand 0yb. Also

suppose that the joint probability density function

fXY(x,y)g(x)h(y), where g(x) is a function only of

xand h(y) is a function only of y. Show that Xand Y

are independent.

V 1 Y 2 c^21 V 1 X 12 c^22 V 1 X 22 pc^2 pV 1 Xp 2

p

MIND-EXPANDING EXERCISES

In the E-book, click on any

term or concept below to

go to that subject.

Bivariate normal

distribution

Conditional mean

Conditional probability

density function

Conditional probability

mass function

Conditional variance

Contour plots

Correlation

Covariance

Independence

Joint probability density

function

Joint probability mass

function

Linear combinations of

random variables

Marginal probability

distribution

Multinomial

distribution

Reproductive property

of the normal distri-

bution

CD MATERIAL

Convolution

Functions of random

variables

Jacobian of a transfor-

mation

Moment generating

function

Uniqueness property of

moment generating

function

Chebyshev’s inequality

IMPORTANT TERMS AND CONCEPTS

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