1. [28 marks] Let X = (X1, X2, ..., Xn) (n 2 2) be i.i.d. random variables having Poisson distribution with density f(x, A) = , I = 0, 1, 2,... a) Show that the product of indicators I(mo}(X) . I(x2 0) (X ) is an unbiased estimator of the parameter 7(1) = e-24. b) Given that T = > X; is complete and minimal sufficient for A, derive i=1 the UMVUE of T()) = e-24. (Hint: you may use part (a)). c) Does the variance of the UMVUE in b) attain the Cramer-Rao bound for the minimal variance of an unbiased estimator of r()) = e-2)? Give reasons for your answer. d) Suppose now that for the same sample, the parameter of interest is h()) = VA. Find the MLE of vX, state its asymptotic distribution and, using this result, suggest a confidence interval for A that asymptotically has a level 1 - a. Explain why h(A) = VA is called "variance stabilising transformation" and why such a transformation it useful. Hint: For any smooth function h(1) : Vn(h(Amie) - h(to)) N(O, (27(10))21-'(No)), 22 I()) = Efax [inf(x, A)])? = EX-azz linf(x, X)]}. e) Let n = 6. Find the uniformly most powerful test of size a = 0.1 for testing Ho : A > 0.25. Justify your