Page 266 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 71 ESTIMATION THEORY

Clearly the m.s. error e depends on c, and it is minimum if

Thus, we conclude that the mas. estimate c of Y is given by

Find the m.s. estimator of a r.v. Y by a function g(X) of the r.v. X.
By Eq. (7.17), the m.s. error is

Since f (x, y) = f (y I x) f (x), we can write

Since the integrands above are positive, the m.s. error e is minimum if the inner integrand,

is minimum for every x. Comparing Eq. (7.58) with Eq. (7.55) (Prob. 7.16), we see that they are the same
form if c is changed to g(x) and f (y) is changed to f (y 1 x). Thus, by the result of Prob. 7.16 [Eq. (7.56)], we
conclude that the m.s. estimate of Y is given by

Hence, the m.s. estimator of Y is

7.18. Find the m.s. error if g(x) = E(Y ( x) is the m.s. estimate of Y,
As we see from Eq. (3.58), the conditional mean E(Y I x) of Y, given that X = x, is a function of x, and
by Eq. (4.39),

SimilarIy, the conditional mean E[g(X, Y) I x] of g(X, Y), given that X = x, is a function of x. It defines,
therefore, the function E[g(X, Y) I XI of the r.v. X. Then

Note that Eq. (7.62) is the generalization of Eq. (7.61). Next, we note that