Introduction to Probability and Statistics for Engineers and Scientists

Problems 139

number of trials that result in outcomei, and show that Cov(N 1 ,N 2 )=−np 1 p 2.

Also explain why it is intuitive that this covariance is negative. (Hint: Fori=

1,...,n, let

Xi=

{

1 if trialiresults in outcome 1

0 if trialidoes not result in outcome 1

Similarly, forj=1,...,n, let

Yj=

{

1 if trialjresults in outcome 2

0 if trialjdoes not result in outcome 2

Argue that

N 1 =

∑n

i= 1

Xi, N 2 =

∑n

j= 1

Yj

Then use Proposition 4.7.2 and Theorem 4.7.4.)

51.In Example 4.5f, compute Cov(Xi,Xj) and use this result to show that Var(X)=1.

52.IfX 1 andX 2 have the same probability distribution function, show that

Cov(X 1 −X 2 ,X 1 +X 2 )= 0

Note that independence is not being assumed.

53.Suppose thatXhas density function

f(x)=e−x, x> 0

Compute the moment generating function ofXand use your result to determine

its mean and variance. Check your answer for the mean by a direct calculation.

54.If the density function ofXis

f(x)=1, 0<x< 1

determineE[etX]. Differentiate to obtainE[Xn]and then check your answer.

55.Suppose thatX is a random variable with mean and variance both equal to 20.

What can be said aboutP{ 0 ≤X≤ 40 }?