Problem 7.5 (a) Let X1,...,Xn be a random sample from the Poisson distribution f(x; ) = e x x! ; x = 0, 1,....Show that

Problem 7.5

(a) Let X1,...,Xn be a random sample from the Poisson distribution f(x; λ) = e−λ λx x! ; x = 0, 1,....Show that X is an unbiased estimator of λ.

(b) Compute the standard error σ(X) and its estimated standard error s(X).

(c) Refer to the data of Problem 1.72. Assuming the data come from a Poisson distribution with parameter λ, use the sample mean to estimate λ and compute the estimated standard error of your estimate.