Consider that X1, X2, Xn are random variables with density f(x|θ) = eθθx/x!, x = 0,1,....
(1) Calculate the Fisher information I(θ) (textbook page 276, Chap- ter 8, 8.5.2 Lemma A).

(2) Find the method of moments estimator θ for θ.
(3) Write down the likelihood of θ.

(4) Find a sufficient statistic for θ.
(5) Find the maximum likelihood estimator θ for θ.

(6) Prove that θ is an unbiased estimator for θ.

(7) Find the variance of θ.

(8) Tell whether θ attaints the Cramer-Rao lower bound.

(9) * Tell whether nI(θ)(θ  θ) for a large n follows a standard
normal distribution.
(10) * Construct an approximate (1  α) confidence interval for θ with Zp denotes the p-th quantile of the standard normal distribution (Note that Zp = Z1p).