Page 297 - Fundamentals of Probability and Statistics for Engineers

280 Fundamentals of Probability and Statistics for Engineers

and
^
lim varf gˆ 0;
n!1
^
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,

" 2 #
1 x m†
2
f x; m; †ˆ exp 2 ; 1 < x < 1: 9:63†
2 † 1=2 2
Estimate parameters 1 ˆ m , and 2 ˆ 2 , 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
1 ˆ M 1 ˆ X;

and
2 ˆ M 2 :

Now,
1 ˆ 1 :

Hence, the first of these moment equations gives
n
1 X
^
1 ˆ X ˆ X j : 9:64†
n
jˆ1
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
n
^
2
^ 2
‡ 2 ˆ M 2 ˆ 1 X X ;
1 j
n
jˆ1
or
n
1 X 2
^
2
2 ˆ M 2 M ˆ X j X† : 9:65†
1
n
jˆ1
This, as we have shown, is a biased estimator for 2 .

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