Introduction to Probability and Statistics for Engineers and Scientists

Problems 289

58.Determine 100(1−α) percent one-sided upper and lower confidence intervals for

θin Problem 57.

59.LetX 1 ,X 2 ,...,Xndenote a sample from a population whose mean valueθis

unknown. Use the results of Example 7.7b to argue that among all unbiased

estimators ofθof the form

∑n

i= 1 λiXi,

∑n

i= 1 λi=1, the one with minimal mean

square error hasλi≡1/n,i=1,...,n.

60.Consider two independent samples from normal populations having the same

varianceσ^2 , of respective sizesnandm. That is,X 1 ,...,XnandY 1 ,...,Ymare

independent samples from normal populations each having varianceσ^2. LetS^2 x

andSy^2 denote the respective sample variances. Thus bothSx^2 andSy^2 are unbiased

estimators ofσ^2. Show by using the results of Example 7.7b along with the fact

that

Var(χk^2 )= 2 k

whereχk^2 is chi-square withkdegrees of freedom, that the minimum mean square

estimator ofσ^2 of the formλSx^2 +(1−λ)Sy^2 is

Sp^2 =

(n−1)Sx^2 +(m−1)Sy^2

n+m− 2

This is called thepooled estimatorofσ^2.

61.Consider two estimators d 1 and d 2 of a parameter θ.IfE[d 1 ]=θ,

Var(d 1 ) =6 andE[d 2 ]=θ+2, Var(d 2 ) = 2, which estimator should be

preferred?

62.Suppose that the number of accidents occurring daily in a certain plant has a

Poisson distribution with an unknown meanλ. Based on previous experience

in similar industrial plants, suppose that a statistician’s initial feelings about the

possible value ofλcan be expressed by an exponential distribution with parameter

1. That is, the prior density is

p(λ)=e−λ,0<λ<∞

Determine the Bayes estimate ofλif there are a total of 83 accidents over the next

10 days. What is the maximum likelihood estimate?

63.The functional lifetimes in hours of computer chips produced by a certain

semiconductor firm are exponentially distributed with mean 1/λ. Suppose that

the prior distribution onλis the gamma distribution with density function

g(x)=

e−xx^2

2

,0<x<∞