Page 269 - Schaum's Outlines - Probability, Random Variables And Random Processes

262 ESTIMATION THEORY [CHAP 7

By Eq. (7.18), the minimum m.s. error estimator of Y in terms of X is
P= E(Y1X)
Now, when X and Y are jointly normal, by Eq. (3.1 08) (Prob. 3.51), we have

Hence, the minimum m.s. error estimator of Y is

Comparing Eq. (7.72) with Eqs. (7.19) and (7.22), we see that for jointly normal r.v.'s the linear m.s. estima-
tor is the minimum m.s. error estimator.

Supplementary Problems

7.25. Let (XI, . . ., X,) be a random sample of X having unknown mean p and variance a2. Show that the
estimator of a2 defined by

where X is the sample mean, is an unbiased estimator of a2. Note that SI2 is often called the sample
variance.

Hint: Show that S12 = -!- S2, and use Eq. (7.29).
n-1

7.26. Let (XI, . . . , X,) be a random sample of X having known mean p and unknown variance a2. Show that the
estimator of a2 defined by

is an unbiased estimator of a2.
Hint: Proceed as in Prob. 7.2.

7.27. Let (XI, . . . , X,) be a random sample of a binomial r.v. X with parameter (m, p), where p is unknown. Show
that the maximum-likelihood estimator of p given by Eq. (7.34) is unbiased.
Hint: Use Eq. (2.38).

7.28. Let (XI, . . . , X,) be a random sample of a Bernoulli r.v. X with pmf f (x; p) = px(l - p)'-", x = 0, 1, where
p, 0 I p I 1, is unknown. Find the maximum-likelihood estimator of p.

Ans. P,, = 1 x Xi =
n i=l
7.29. The values of a random sample, 2.9, 0.5, 1.7, 4.3, and 3.2, are obtained from a r.v. X that is uniformly
distributed over the unknown interval (a, b). Find the maximum-likelihood estimates of a and b.
Ans. 2,, = min xi = 0.5, hML = max xi = 4.3
1 I