Page 262 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 73 ESTIMATION THEORY

Thus,

In order to find the values of ,u and a maximizing the above, we compute

Equating these equations to zero, we get

Solving for jML and &ML, the maximum-likelihood estimates of p and a2 are given, respectively, by

Hence, the maximum-likelihood estimators of p and a2 are given, respectively, by

BAYES' ESTIMATION
7.11. Let (XI, . . . , X,) be the random sample of a Bernoulli r.v. X with pmf given by [Eq. (2.32)]
f (x; P) = PX(l - P)' -" x=o, 1 (7.43)
where p, 0 < p I is unknown. Assume that p is a uniform r.v. over (0, 1). Find the Bayes'
1,
estimator of p.
The prior pdf of p is the uniform pdf; that is,

The posterior pdf of p is given by

Then, by Eq. (7.1 2),

where m = z;=, xi, and by Eq. (7.1 3),

Now, from calculus, for integers m and k, we have
m! k!
[pm(l - dp = --
(m + k $- I)!