Suppose that X1,..., Xn are iid Poisson(). Find the best unbiased estimator of (a) e-, the probability

Question:

Suppose that X1,..., Xn are iid Poisson(λ). Find the best unbiased estimator of
(a) e-λ, the probability that X = 0.
(b) λe-λ, the probability that X = 1.
(c) For the best unbiased estimators of parts (a) and (b), calculate the asymptotic relative efficiency with respect to the MLE. Which estimators do you prefer? Why?
(d) A preliminary test of a possible carcinogenic compound can be performed by measuring the mutation rate of microorganisms exposed to the compound. An experimenter places the compound in 15 petri dishes and records the following number of mutant colonies:
10, 7, 8, 13, 8, 9, 5, 7, 6, 8, 3, 6, 6, 3, 5.
Estimate e-λ, the probability that no mutant colonies emerge, and λe-λ, the probability that one mutant colony will emerge. Calculate both the best unbiased estimator and the MLE.