5. The number of customers that enter the corner grocery store during the noon hour has a Poisson distribution, i.e., fðz; lÞ ¼ el lz z! If0;1;2;3;...g ðzÞ:

Assume that ð Þ X1; X2; ... ; Xn 0 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 l. Is X the MVUE for estimating l? Why or why not?

c. Use the CRLB attainment theorem to derive the MVUE for estimating l. Suppose n ¼ 100 and P100 i¼1 xi ¼ 283. Estimate l using the MVUE.

d. Is X a member of the CAN class of estimators? Is X asymptotically efficient?

e. Define the CRLB for estimating P(z ¼ 0) ¼ el

. Does there exist an unbiased estimator of el that achieves the CRLB? Why or why not?