Page 292 - Applied Statistics And Probability For Engineers

PQ220 6234F.CD(07) 5/15/02 4:31 PM Page 4 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark F

7-4

Therefore, the desired distribution is

f 1x 1 , x 2 , p , x n , 2
f 1 0 x , x , p , x 2
n
1
2
f 1x , x , p , x 2
2
n
1

We define the Bayes estimator of as the value that corresponds to the mean of the poste-
rior distribution f 1 0 x , x , p , x 2.
n
2
1
Sometimes, the mean of the posterior distribution of can be determined easily. As a
function of , f 1 0 x , p , x 2 is a probability density function and x , p , x n are just con-
1
1
n
stants. Because enters into f 1 0 x , p , x 2 only through f 1x , p , x , 2 if f 1x , p , x , 2,
1
1
n
n
n
1
as a function of is recognized as a well-known probability function, the posterior mean of
can be deduced from the well-known distribution without integration or even calculation of
f 1x , p , x 2.
n
1
EXAMPLE S7-2 Let X 1 , X , p , X be a random sample from the normal distribution with mean and variance
2
n
2
2
, where is unknown and is known. Assume that the prior distribution for is normal
2
with mean and variance 0 ; that is
0
1 2 2 1 2 2 2
f 1 2 e 1 0 2 12 0 2 2 e 1 2 0 0 2 12 0 2
12
0 12
0
The joint probability distribution of the sample is
1 2 n 2
f 1x , x , p , x 0 2 2 n 2 e 11 2 2 a 1x i 2
1
2
i 1
n
12
2
1 11 2 21 ax i 2 a x i n 2 2
2
2
2 n 2 e
12
2
Thus, the joint probability distribution of the sample and is
1 2 2 2 2 2 2 2 2 2
f 1x , x , p , x , 2 2 n 2 e 11 22311 0 n 2 12 0 0 2a x i 2 a x i 0 0 4
n
2
1
12
2 12
0
1 1 0 x
2
11 22 ca 2 2 b 2 a 2 b d
2
2
2
e 0 n 0 n h 1x , p , x , , , 0 2
1
n
1
0
Upon completing the square in the exponent
2 2 2
1 1 1 n2 0 x 0
2
11 22 a 2 2 b c a 2 2 2 2 bd
2
2
f 1x , x , p , x , 2 e 0 n 0 n 0 n h 1x , p , x , , , 0 2
2
n
n
1
2
0
1
2 2 2 2
where h (x , p , x , , , 0 ) is a function of the observed values, , , and 0 .
1
n
0
0
i
Now, because f(x , p , x ) does not depend on ,
n
1
2
2
1 1 1 n2 0 0 x
11 22 a 2 2 b c a 2 2 bd
2
2
2
f 1 0 x , p , x 2 e 0 n 0 n h 1x , p , x , , , 0 2
n
1
3
n
1
0