Consider the simple Poisson model based on f (x; ) = xe /x!, >0, x = 0, 1, 2, . . . (a) Derive the

Consider the simple Poisson model based on f (x; θ) = θxeθ /x!, θ>0, x = 0, 1, 2, . . .

(a) Derive the Cramer–Rao lower bound for unbiased estimators of θ.

(b) Explain why X = 1n

ni

=1 Xi is an unbiased and fully efficient estimator of θ.

(c) Derive the posterior distribution when π(θ) is Gamma(α, β).