Page 265 - Schaum's Outlines - Probability, Random Variables And Random Processes

ESTIMATION THEORY [CHAP 7

and the Bayes' estimator of p is

7.14. Let (XI, . . ., X,) be a random sample of a r.v. X with pdf f(x; 8), where 8 is an unknown
parameter. The statistics L and U determine a 10q1 - a) percent confidence interval (L, U) for
the parameter 8 if
P(L<O<U)>l-a O<a<l (7.51)
and 1 - a is called the conjdence coefficient. Find L and U if X is a normal r.v. with known
variance a2 and mean p is an unknown parameter.
If X = N(p; a2), then

is a standard normal r.v., and hence for a given a we can find a number za12 from Table A (Appendix A)
such that

.
For example, if 1 - a = 0.95, then z,,, = z,.,,, = 1.96, and if 1 - a = 0.9, then za12 = z ~ = 1.645. Now,
~
~
recalling that o > 0, we have the following equivalent inequality relationships;
x-p
-.=a/, < -
a/&

7.15. Consider a normal r.v. with variance 1.66 and unknown mean p. Find the 95 percent confidence
interval for the mean based on a random sample of size 10.
As shown in Prob. 7.14, for 1 - a = 0.95, we have za12 = z,,,,, = 1.96 and
za12(o/&) = 1.96(@/fi) = 0.8
Thus, by Eq. (7.54), the 95 percent confidence interval for p is
(K - 0.8, X + 0.8)

MEAN SQUARE ESTIMATION
7.16. Find the m.s. estimate of a r.v. Y by a constant c.
By Eq. (7.17), the m.s. error is