Page 270 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 71 ESTIMATION THEORY 263

In analyzing the flow of traffic through a drive-in bank, the times (in minutes) between arrivals of 10
customers are recorded as 3.2, 2.1, 5.3, 4.2, 1.2, 2.8, 6.4, 1.5, 1.9, and 3.0. Assuming that the interarrival time
is an exponential r.v. with parameter I, find the maximum likelihood estimate of I.

1
Ans. I,, = -
3.16
Let (XI, . . . , X,) be a random sample of a normal r.v. X with known mean ,u and unknown variance a2.
Find the maximum likelihood estimator of a2.
1 --
Ans. SML2 = - x (Xi -
n i=l
Let (XI, . . ., X,) be the random sample of a normal r.v. X with mean p and variance a2, where p is
unknown. Assume that p is itself to be a normal r.v. with mean p, and variance aI2. Find the Bayes'
estimate of p.

Let (XI, . . . , X,) be the random sample of a normal r.v. X with variance 100 and unknown p. What sample
size n is required such that the width of 95 percent confidence interval is 5?
Ans. n = 62

Find a constant a such that if Y is estimated by ax, the m.s. error is minimum, and also find the minimum
m.s. error em.
Ans. a = E(XY)/E(X2) em = E(Y2) - [E(X Y)I2/[E(X)l2

Derive Eqs. (7.25) and (7.26).
Hint: Proceed as in Prob. 7.20.