The operating life of a small electric motor manufactured by the AJAX Electric Co. can be represented as a random variable having a probability density given as

\(Z \sim f(z ; \Theta)=\frac{1}{6 \Theta^{4}} z^{3} e^{-z / \Theta} I_{(0, \infty)}(z)\)

where \(\Theta \in \Omega=(0, \infty), \mathrm{E}(Z)=4 \Theta, \operatorname{var}(Z)=4 \Theta^{2}\), and \(z\) is measured in thousand of hours. A random sample \(\left(X_{1}, \ldots, X_{100}ight)\) of the operating lives of 100 electric motors has an outcome that is summarized as \(\bar{X}=7.65\) and \(s^{2}=\sum_{i=1}^{n}\left(x_{i}-\bar{x}ight)^{2} / 100=1.73\).

a. Define a minimal, complete sufficient statistic for estimating the expected operating life of the electric motors produced by the AJAX Co.

b. Define the MVUE for estimating \(E(Z)=4 \Theta\). Justify the MVUE property of your estimator. Generate an estimate of \(\mathrm{E}(Z)\) using the MVUE.

c. Is the MVUE estimator a consistent estimator of \(\mathrm{E}(Z)=4 \Theta\) ? Why or why not?

d. Does the variance of the estimator you defined in

(b) attain the Cramer-Rao Lower Bound? (The CRLB regularity conditions hold for the joint density of the random sample. Furthermore, the alternative form of the CRLB, expressed in terms of secondorder derivatives, applies in this case if you want to use it).