The number of customers that enter the corner grocery store during the noon hour has a Poisson distribution, i.e.,

\(f(z ; \lambda)=\frac{e^{-\lambda} \lambda^{z}}{z !} I_{\{0,1,2,3, \ldots\}}(z)\).

Assume that \(\left(X_{1}, X_{2}, \ldots, X_{n}ight)^{\prime}\) is a random sample from this Poisson population distribution.

a. Show that the Cramer-Rao lower bound regularity conditions hold for the joint density of the random sample.

b. Derive the CRLB for unbiased estimation of the parameter \(\lambda\). Is \(\bar{X}\) the MVUE for estimating \(\lambda\) ? Why or why not?

c. Use the CRLB attainment theorem to derive the MVUE for estimating \(\lambda\). Suppose \(n=100\) and \(\sum_{i=1}^{100} x_{i}=283\). Estimate \(\lambda\) using the MVUE.

d. Is \(\bar{X}\) a member of the CAN class of estimators? Is \(\bar{X}\) asymptotically efficient?

e. Define the CRLB for estimating \(P(z=0)=e^{-\lambda}\). Does there exist an unbiased estimator of \(e^{-\lambda}\) that achieves the CRLB? Why or why not?