Fundamentals of Probability and Statistics for Engineers

Ex ample 9. 12. Suppose that population X has a uniform distribution over the
range (0, ) and we wish to estimate parameter from a sample of size n.
The density function of X is

and the first moment is

It follows from the method of moments that, on letting we obtain

Upon little reflection, the validity of this estimator is somewhat questionable
because, by definition, all values assumed by X are supposed to lie within
interval (0, ). However, we see from Equation (9.75) that it is possible that
some of the samples are greater than. Intuitively, a better estimator might be

where X(n) is the nth-order statistic. As we will see, this would be the outcome
following the method of maximum likelihood, to be discussed in the next
section.
Since the method of moments requires only i, the moments of population X,
the knowledge of its pdf is not necessary. This advantage is demonstrated in
Example 9.13.

Ex ample 9. 13. Problem: consider measuring the length r of an object with use
of a sensing instrument. Owing to inherent inaccuracies in the instrument, what
is actually measured is X, as shown in Figure 9.3, where X 1 and X 2 are
identically and normally distributed with mean zero and unknown variance

(^2). Determine a moment estimator for r (^2) on the basis of a sample of size
n from X.
Answer: now, random variable X is
The pdf of X with unknown parameters and^2 can be found by using
techniques developed in Chapter 5. It is, however, unnecessary here since some
Parameter Estimation 283

 

f…x;†ˆ

1



; for 0 x;

0 ; elsewhere;

8

<

… 9 : 73 †

1 ˆ



2

: … 9 : 74 †

1 ˆX,

^ˆ 2 Xˆ^2

n

Xn

jˆ 1

Xj: … 9 : 75 †



^

^ˆX…n†; … 9 : 76 †

 ^ ˆ

Xˆ‰…r‡X 1 †^2 ‡X 22 Š^1 =^2 : … 9 : 77 †

 

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