Fundamentals of Probability and Statistics for Engineers

and

and hence is consistent.

Example 9.9.Problem: let us select the normal distribution as a model for the
percentage yield discussed in Chapter 8; that is,

Estimate parameters ,and , based on the 200 sample values given
in Table 8.1, page 249.
Answer: following the method of moments, we need two moment equations,
and the most convenient ones are obviously

and

Now,

Hence, the first of these moment equations gives

The properties of this estimator have already been discussed in Example 9.2. It
is unbiased and has minimum variance among all unbiased estimators for m.
We see that the method of moments produces desirable results in this case.
The second moment equation gives

or

This, as we have shown, is a biased estimator for^2.

280 Fundamentals of Probability and Statistics for Engineers

lim

n!1

varf^gˆ 0 ;

^

f…x;m;^2 †ˆ

1

… 2 †^1 =^2 

exp 

…xm†^2

2 ^2

"

; 1<x< 1 : … 9 : 63 †

 1 ˆm  2 ˆ^2

1 ˆM 1 ˆX;

2 ˆM 2 :

1 ˆ 1 :

^ 1 ˆXˆ^1

n

Xn

jˆ 1

Xj: … 9 : 64 †

^^21 ‡^ 2 ˆM 2 ˆ^1

n

Xn

jˆ 1

Xj^2 ;

^ 2 ˆM 2 M^2

1 ˆ

1

n

n

jˆ 1

…XjX†^2 : … 9 : 65 †



X