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

\(Z \sim f(z ; \theta)=\frac{1}{\theta} e^{-z / \theta} I_{(0, \infty)}(z)\), where \(z=\) the number of hours the math coprocessor functions until failure, measured in thousands of hours.

A random sample, \(X=\left(X_{1}, \ldots, X_{n}ight)\), 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 \(\bar{x}=28.7\).

a. Define a minimal sufficient statistic for \(f(\mathbf{x} ; \theta)\), the joint density of the random sample.

b. Define a complete sufficient statistic for \(f(\mathbf{x} ; \theta)\).

c. Define the MVUE for \(\mathrm{E}(Z)=\theta\) if it exists. Estimate \(\theta\).

d. Define the MVUE for \(\operatorname{var}(Z)=\theta^{2}\) if it exists. Estimate \(\theta^{2}\).

e. Define the MVUE for \(E\left(Z^{2}ight)=2 \theta^{2}\) if it exists.

Estimate \(2 \theta^{2}\).

f. Define the MVUE for \(q(\theta)_{(3 \times 1)}=\left[\begin{array}{c}\theta \\ \theta^{2} \\ 2 \theta^{2}\end{array}ight]\) if it exists.

g. Is the second sample moment about the origin, i.e., \(M^{\prime}{ }_{2}=\sum_{i=1}^{n} X_{i}^{2} / n\), the MVUE for \(\mathrm{E}\left(Z^{2}ight)\) ? h. Is the sample variance, \(S^{2}\), the MVUE for \(\operatorname{var}(Z)\) ?

i. Suppose we want the MVUE for \(F(b)=P(z \leq b)=\) \(1-e^{-b / \theta}\), where \(F(b)\) is the probability that the coprocessor fails before \(1,000 b\) hours of use. It can be shown that

\(t(\mathbf{X})=1-\left(1-\frac{b}{\left(\sum_{i=1}^{n} X_{i}ight)}ight)^{n-1} I_{[b, \infty)}\left(\sum_{i=1}^{n} X_{i}ight)\)

is such that \(\mathrm{E}(t(\mathbf{X}))=1-e^{-b / \theta}\). Is \(t(\mathbf{X})\) the MVUE for \(P(z \leq b)\) ? Why or why not? Estimate \(P(z \leq 20)\).

(j) Is \(t_{*}(\mathbf{X})=1-e^{-b / \bar{X}}\) a MVUE for \(F(b)\) ? Is \(t_{*}(\mathbf{X})\) a consistent estimator of \(F(b)\) ?