Let X1,...,Xn be iid Poisson(A), and let and S2 denote the sample mean and variance, respectively.

Question:

Let X1,...,Xn be iid Poisson(A), and let and S2 denote the sample mean and variance, respectively. We now complete Example 7.3.8 in a different way. There we used the Cramer-Rao Bound; now we use completeness.
(a) Prove that is the best unbiased estimator of λ without using the Cramer-Rao Theorem.
(b) Prove the rather remarkable identity E(S2|) = , and use it to explicitly demonstrate that VarS2 > Var.
(c) Using completeness, can a general theorem be formulated for which the identity in part (b) is a special case?