Introduction to Probability and Statistics for Engineers and Scientists

Problems 137

42.Argue that for any random variableX

E[X^2 ]≥(E[X])^2

When does one have equality?

43.A random variableX, which represents the weight (in ounces) of an article, has

density function given byf(z),

f(z)=







(z−8) for 8≤z≤ 9

(10−z) for 9<z≤ 10

0 otherwise

(a) Calculate the mean and variance of the random variableX.

(b)The manufacturer sells the article for a fixed price of $2.00. He guarantees

to refund the purchase money to any customer who finds the weight of his

article to be less than 8.25 oz. His cost of production is related to the weight

of the article by the relationx/15+.35. Find the expected profit per article.

44.Suppose that the Rockwell hardnessXand abrasion lossYof a specimen (coded

data) have a joint density given by

fXY(u,v)=

{

u+v for 0≤u,v≤ 1

0 otherwise

(a) Find the marginal densities ofXandY.

(b)FindE(X) and Var(X).

45.A product is classified according to the number of defects it contains and the

factory that produces it. LetX 1 andX 2 be the random variables that represent

the number of defects per unit (taking on possible values of 0, 1, 2, or 3) and the

factory number (taking on possible values 1 or 2), respectively. The entries in the

table represent the joint possibility mass function of a randomly chosen product.

X 2

X
 1 12

0 18 161

1 161 161

(^216318)
(^31814)
(a) Find the marginal probability distributions ofX 1 andX 2.
(b)FindE[(X 1 )],E[(X 2 )], Var(X 1 ), Var(X 2 ), and Cov(X 1 ,X 2 ).