Applied Statistics and Probability for Engineers

When an estimator is unbiased, the bias is zero; that is,

EXAMPLE 7-1 Suppose that Xis a random variable with mean and variance. Let be a

random sample of size nfrom the population represented by X. Show that the sample mean

and sample variance are unbiased estimators of and , respectively.

First consider the sample mean. In Equation 5.40a in Chapter 5, we showed that

Therefore, the sample mean is an unbiased estimator of the population mean .

Now consider the sample variance. We have

The last equality follows from Equation 5-37 in Chapter 5. However, since

and we have

Therefore, the sample variance is an unbiased estimator of the population variance

Although is unbiased for ^2 , Sis a biased estimator of . For large samples, the bias is very

small. However, there are good reasons for using Sas an estimator of  in samples from nor-

mal distributions, as we will see in the next three chapters when are discuss confidence

intervals and hypothesis testing.

S^2

S^2 ^2.

^2



1

n 1

1 n^2 n^2 n^2 ^22

E 1 S^22 

1

n 1

ca

n

i 1

1 ^2 ^22 n 1 ^2 ^2 n2d

E 1 X^22 ^2 ^2 n,

E 1 X^2 i 2 ^2 ^2



1

n 1

ca

n

i 1

E 1 X^2 i 2 nE 1 X^2 2d



1

n 1

E (^) a
n
i 1
1 X^2 iX^2  2 XXi 2 
1
n 1
E aa
n
i 1
X^2 inX^2 b
E 1 S^22 E £
a
n
i 1
1 XiX 22
n 1
§
1
n 1
E (^) a
n
i 1
1 XiX 22
X
E 1 X 2 .
S^2 ^2
X
^2 X 1 , X 2 ,p, Xn
E 1 ˆ 2 0.
The point estimator is an unbiased estimatorfor the parameter if
(7-1)
If the estimator is not unbiased, then the difference
(7-2)
is called the biasof the estimator .ˆ
E 1 ˆ 2 
E 1 ˆ 2 
ˆ
Definition
7-2 GENERAL CONCEPTS OF POINT ESTIMATION 223
c07.qxd 5/15/02 10:18 M Page 223 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: