Page 260 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 71 ESTIMATION THEORY 253

If X is uniformly distributed over (0, a), then from Eqs. (2.44), (2.45), and (4.98) of Prob. 4.30, the pdf of
Z = max(X,, . . . , X,) is

Thus

and lim E(A) = a
n+ w

Next,

and lim Var(A) = 0
n-* w
Thus, by Eqs. (7.7) and (7.8), A is a consistent estimator of parameter a.

MAXIMUM-LIKELIHOOD ESTIMATION
7.7. Let (XI, . . . , X,) be a random sample of a binomial r.v. X with parameters (m, p), where m is
assumed to be known and p unknown. Determine the maximum-likelihood estimator of p.
The likelihood function is given by [Eq. (2.36)]

Taking the natural logarithm of the above expression, we get

where

and

Setting dun LCp)]/dp = 0, the maximum-likelihood estimate jML of' p is obtained as

Hence, the maximum-likelihood estimator of p is given by
I n I

7.8. Let (XI, . . . , X,) be a random sample of a Poisson r.v. with unknown parameter A. Determine the
maximum-likelihood estimator of A.