Applied Statistics and Probability for Engineers

7-4

Therefore, the desired distribution is

We d efine the Bayes estimatorof as the value that corresponds to the mean of the poste-

rior distribution

Sometimes, the mean of the posterior distribution of can be determined easily. As a

function of , is a probability density function and are just con-

stants. Because enters into only through if

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

EXAMPLE S7-2 Let X 1 , X 2 , p, Xnbe a random sample from the normal distribution with mean and variance

^2 , where is unknown and ^2 is known. Assume that the prior distribution for is normal

with mean  0 and variance ; that is

The joint probability distribution of the sample is

Thus, the joint probability distribution of the sample and is

Upon completing the square in the exponent

where hi(x 1 , p, xn, ^2 ,  0 , ) is a function of the observed values, ^2 ,  0 , and.

Now, because f(x 1 , p, xn) does not depend on ,

f 1  0 x 1 ,p, xn 2 e

 (^11)  22 a^1
02
1
^2 nb^ c
(^2)  a^1 ^2 n^2 ^0 ^20 x
^20 ^2 n bd^ h
31 x 1 ,p, xn, 
(^2) , 
0 , 
2
02
^20 ^20
f 1 x 1 , x 2 ,p, xn,  2 e
 (^11)  22 a^1
02
1
^2 nb^ c
(^2) a^1 
(^2) n 2  0
^20 ^2 n^
(^)
x^20
^20 ^2 nb
d
2
(^) h 21 x 1 ,p, xn, ^2 ,  0 , ^202
e
 (^11)  22 ca^1
02
1
^2 nb^ 
(^2)  2 a^0
 02
x
^2 nb^ d^ h 11 x 1 ,p, xn, ^2 ,  0 , ^202
f 1 x 1 , x 2 ,p, xn,  2 
1
12
^22 n^212
 0
e^11 ^22311 
(^20) n (^22)  (^2)  12  0  (^20 2) a xi (^22)  a x (^2) i (^2)  (^20)  (^204)

1
12
^22 n^2
e^11 ^2 
(^221) ax (^2) i 2 axi n (^22)
f 1 x 1 , x 2 ,p, xn 0  2 
1
12
^22 n^2
e^11 ^2 
(^22) an
i 11 xi^2
2
f 1  2 
1
12
 0
e^1 ^02
(^2)  12  (^202)

1
12
^20
e^1 
(^2)  2  0   (^202)  12  (^202)
^20
f 1 x 1 ,p, xn 2.
f 1  0 x 1 ,p, xn 2 f 1 x 1 ,p, xn,  2 f 1 x 1 ,p, xn,  2 ,
f 1  0 x 1 ,p, xn 2 x 1 ,p, xn
f 1  0 x 1 , x 2 ,p, xn 2.

f 1  0 x 1 , x 2 ,p, xn 2 
f 1 x 1 , x 2 ,p, xn,  2
f 1 x 1 , x 2 ,p, xn 2
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