please resolve

4. (") a) Let X be a Poisson distributed random variable with parameter A > 0 : P(X = () = 247414,4 = 0, 1, ... The discrete random variable Z with A(Z = k) = P(X = KX > Oj:k = 1, 2... . has then the so-called called zero-truncated Poisson distribution. a) Find the expected value Ex(2); b) Calculate the Fisher information quantities Ix (X) and Iz (A) (in a sample of size n = 1). c) Intuitively, Z is a truncated version of X and due to truncation, information loss is expected. Both information quantities in b) depend on A. Examine their behaviour when A 3 0 is small and when A -+ co. Comment on whether or not the intuition is confirmed. b) For a sample of n Lid. observations X], X3...... X,, from N(0, 8") distribution, argue that the UMVUE of 0 3 0 is I(n/2) V 21((m + 1)/2) S where S' - 1 EL, X7. Clearly outline each step of your argument. Hint: You may need the density of xa distribution: (x(?) = people ,x>0. You are not asked to derive this