Page 263 - Schaum's Outlines - Probability, Random Variables And Random Processes

256 ESTIMATION THEORY [CHAP 7

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

and by Eqs. (7.15) and (7.44),

- 1 - py-m dp
- (n + 1)' +
m!(n - m)!

Hence, by Eq. (7.1 6), the Bayes' estimator of p is
1
PB = E(pIX,, ..., X,) =-
n + 2 (:*xi + I)
7.12. Let (XI, .. ., X,) be a random sample of an exponential r.v. X with unknown parameter A.
Assume that 3, is itself to be an exponential rev. with parameter a. Find the Bayes' estimator of 1.
The assumed prior pdf of 1 is [Eq. (2.48)J
f (4 = i0 a,l>O
ae -""
otherwise

Now
where m = El=, xi . Then, by Eqs. (7.1 2) and (7.13),

By Eq. (7.14), the posterior pdf of 1 is given by

Thus, by Eq. (7.15), the Bayes' estimate of 1 is

AB = E(AI xl, ..., x,) = i'
lf(Alxl, .. ., x,) dl