Introduction to Probability and Statistics for Engineers and Scientists

274 Chapter 7: Parameter Estimation

Now it can be shown that for integral valuesmandr

∫ 1

0

θm(1−θ)rdθ=

m!r!

(m+r+1)!

(7.8.1)

Hence, upon lettingx=

∑n

i= 1 xi

f(θ|x 1 ,...,xn)=

(n+1)!θx(1−θ)n−x

x!(n−x)!

(7.8.2)

Therefore,

E[θ|x 1 ,...,xn]=

(n+1)!

x!(n−x)!

∫ 1

0

θ^1 +x(1−θ)n−xdθ

=

(n+1)!

x!(n−x)!

(1+x)!(n−x)!

(n+2)!

from Equation 7.8.1

=

x+ 1

n+ 2

Thus, the Bayes estimator is given by

E[θ|X 1 ,...,Xn]=

∑n

i= 1

Xi+ 1

n+ 2

As an illustration, if 10 independent trials, each of which results in a success with probability
θ, result in 6 successes, then assuming a uniform (0, 1) prior distribution onθ, the Bayes
estimator ofθis 7/12 (as opposed, for instance, to the maximum likelihood estimator
of 6/10). ■

REMARK

The conditional distribution ofθgiven thatXi=xi,i=1,...,n, whose density function
is given by Equation 7.8.2, is called the beta distribution with parameters

∑n
i= 1 xi+1,
n−

∑n

i= 1 xi+1. ■

EXAMPLE 7.8b SupposeX 1 ,...,Xnare independent normal random variables, each having
unknown meanθand known varianceσ 02 .Ifθis itself selected from a normal population
having known meanμand known varianceσ^2 , what is the Bayes estimator ofθ?

SOLUTION In order to determineE[θ|X 1 ,...,Xn], the Bayes estimator, we need first
determine the conditional density ofθgiven the values ofX 1 ,...,Xn. Now

f(θ|x 1 ,...,xn)=

f(x 1 ,...,xn|θ)p(θ)

f(x 1 ,...,xn)