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 }?