Fundamentals of Probability and Statistics for Engineers

whereX is the sample mean based on a sample of size n. The choice of 1 is
intuitively obvious since , and the choice of 2 is based on a prior
probability argument that is not our concern at this point.
Since

and

we have

and

We see from the above that, although 2 is a biased estimator, its variance is
smaller than that of 1 , particularly when n is of a moderate value. This is

(a) (b)

Figure 9.2 Probability density functions of 1 and 2

Parameter Estimation 273

^

EfXgˆ ^

EfXgˆ;

^2 Xˆ

… 1 †

n

Ef^ 1 gˆ;

Ef^ 2 gˆ

n‡ 1

n‡ 2

;

9

=

;

… 9 : 45 †

^2 ^ 1 ˆ

… 1 †

n

;

^2 ^ 2 ˆ

n^2

…n‡ 2 †^2

^2 Xˆ

n… 1 †

…n‡ 2 †^2

:

9

>>>

=

>>

>;

… 9 : 46 †

^

^

θ θ

fΘ∧ 2 (∧θ 2 )

fΘ∧ 1 (∧θ 1 )

fΘ∧ 2 (∧θ 2 )

fΘ∧ 1 (∧θ 1 )

θ 1 ,θ 2

∧∧

θ∧∧ 1 ,θ 2

^ ^