Math 370 HW 5 - Homework 5 Solution 4-5, 4-6, 4-28 (d),...

1
Homework 5 Solution
4-5, 4-6, 4-28 (d), 4-29 (e), 4-37 (a, b, d), 4-40 (a, d)
4-5.
E
 
 
X E
X
n n
E X
n
n
i
i
n
i
i
n
2





















 




 1
1
1 1
 
,
X
1
and
X
2
are unbiased estimators of  .
The variances are V
 
X
1
2
2n


and V
 
X
2
2
n


; compare the MSE (variance in this case),
MSE
MSE
n
n
n
n
(

)
(

)
/
/




1
2
2
2
2
2
1
2
  
Because both estimators are unbiased, examination of the variances would conclude that
X
1
is the “better” estimator with the smaller variance.
4-6. E
 
 
          ) 9 (
9
1
)) ( 9 (
9
1
) ( ) ( ) (
9
1
ˆ
9 2 1 1
X E X E X E X E 
E
 
 
     2 ] 2 3 [
2
1
) 2 ( ) ( ) 3 (
2
1
ˆ
4 6 1 2
       X E X E X E
a)


1
is unbiased estimates and because the expected values of these statistics are
equivalent to the true mean,  ., but 

2
is biased estimates of 
b) V
 
 
2 2
9 2 1
2
9 2 1
1
9
1
) 9 (
81
1
) X ( V ) X ( V ) X ( V
9
1
9