Page 212 - Applied Statistics And Probability For Engineers

c05.qxd 5/13/02 1:50 PM Page 188 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

188 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

MIND-EXPANDING EXERCISES

5-122. Show that if X 1 , X 2 , p , X p are independent, 5-124. Suppose that the joint probability function of
continuous random variables, P(X 1 A 1 , X 2 A 2 , p , the continuous random variables X and Y is constant on
X p A p ) P(X 1 A 1 )P(X 2 A 2 ) p P(X p A p ) for any the rectangle 0 x a, 0 y b. Show that X and Y
are independent.
regions A 1 , A 2 , p , A p in the range of X 1 , X 2 , p , X p
respectively. 5-125. Suppose that the range of the continuous
5-123. Show that if X 1 , X 2 , p , X p are independent variables X and Y is 0 x a and 0 y b. Also
random variables and Y c 1 X 1 c 2 X 2 p c p X p , suppose that the joint probability density function
f XY (x, y) g(x)h( y), where g(x) is a function only of
2
2
2
V1Y 2 c 1 V1X 1 2 c 2 V1X 2 2 p c p V1X p 2
x and h( y) is a function only of y. Show that X and Y
are independent.
You can assume that the random variables are continuous.

IMPORTANT TERMS AND CONCEPTS
In the E-book, click on any Correlation Multinomial Moment generating
term or concept below to Covariance distribution function
go to that subject. Independence Reproductive property Uniqueness property of
Bivariate normal Joint probability density of the normal distri- moment generating
distribution function bution function
Conditional mean Joint probability mass Chebyshev’s inequality
Conditional probability function CD MATERIAL
density function Linear combinations of Convolution
Conditional probability random variables Functions of random
mass function Marginal probability variables
Conditional variance distribution Jacobian of a transfor-
Contour plots mation