Page 264 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 71 ESTIMATION THEORY

and the Bayes' estimator of rZ is

7.13. Let (XI, . . . , X,) be a random sample of a normal r.v. X with unknown mean p and variance 1.
Assume that p is itself to be a normal r.v. with mean 0 and variance 1. Find the Bayes' estimator
of p.
The assumed prior pdf of p is

Then by Eq. (7.12),

Then, by Eq. (7.14), the posterior pdf of p is given by

where C = C(xl, . . . , xn) is independent of p. However, Eq. (7.48) is just the pdf of a normal r.v. with mean

and variance
1
n+l
Hence, the conditional distribution of p given x,, . . . , x, is the normal distribution with mean

and variance
1
n+l
Thus, the Bayes' estimate of p is given by