1. The operating life of a certain type of math coprocessor installed in a personal computer can be represented as the outcome of a random variable having an exponential density function, as Z  fðz; yÞ ¼ 1 y ez=y Ið0;1Þ ðzÞ;

where z ¼ the number of hours the math coprocessor functions until failure, measured in thousands of hours.

A random sample, X ¼ (X1,...,Xn), of the operating lives of 200 coprocessors is taken, where the objective being is to estimate a number of characteristics of the operating life distribution of the coprocessors. The outcome of the sample mean was x ¼ 28:7.

a. Define a minimal sufficient statistic for f(x;y), the joint density of the random sample.

b. Define a complete sufficient statistic for f(x; y).

c. Define the MVUE for E(Z) ¼ y if it exists. Estimate y.

d. Define the MVUE for var(Z) ¼ y2 if it exists.

Estimate y2

.

e. Define the MVUE for E(Z2

) ¼ 2y2 if it exists.

Estimate 2y2

.

f. Define the MVUE for qð Þy ð Þ 31 ¼

y y2 2y2 2

4 3

5 if it exists.

Estimate q(y).

g. Is the second sample moment about the origin, i.e., M0 2 ¼ Pn i¼1 X2 i =n, the MVUE for E(Z2

)?

h. Is the sample variance, S2

, the MVUE for var(Z)?

i. Suppose we want the MVUE for F

(b) ¼ P(z

b) ¼

1eb/y

, where F

(b) is the probability that the coprocessor fails before 1,000 b hours of use. It can be shown that tðXÞ ¼ 1  1  b Pn i¼1 Xi

  !n1 I½b;1Þ S n

i¼1 Xi



is such that E(t(X)) ¼ 1eb/y

. Is t(X) theMVUE for P(z b)?

Why or why not? Estimate P(z 20).

(j) Is t ðXÞ ¼ 1  eb=X a MVUE for F(b)? Is t*(X) a consistent estimator of F(b)?