Question

(a)Suppose X ∼ Poisson(λ) and Y ∼ Poisson(γ) are independent, prove that X + Y ∼ Poisson(λ + γ).

(b)Let X1, . . . , Xn be an iid random sample from Poisson(λ), provide a sufficient statistic for λ and justify your answer.

(c)Under the setting of part (b), show λb = 1 n Pn i=1 Xi is consistent estimator of λ.

(d)Use the Central Limit Theorem to find an asymptotic normal distribution for λb defined in part (c), justify your answer.

(e)Suppose γ is a random variable with Exp(θ) distribution. Conditioning on γ, Y ∼ Poisson(γ). Provide the marginal mean and variance of Y .

Answer 1

(a) P(Z = X+Y = z) = P(X = 1, Y = 2 - ) T=0 P(X = x) P(Y = 2 – ) (since X and Y are independent) (2) TI e-AT 2 2! e-(+)(a + 6-7h -1 (2-2)! e-(A+) z! 2 ISO = - 2! - 2=0,1,2,... Hence Z ~ Poisson(+7). (6) The joint pmf of X1, X2, ..., Xn is e-ti f1(21, 12, ..., In) = 1 2;)= 11 = gu(7)h(x1, 12,..., Mn); where, g1(T) = e-nyni, h(21, 22, ..., In) = e- nE="i e-niqni IT-10;! 11-12;! i=1 Thi! Hence using Neyman – Fisher Factorization theorem, X = is sufficient statistic for X.

(c) E(X;) = 1 and 11 Var(X) = Var(X;) (since X's are independent) = A +0 as n +0. Hence X is consistent estimator of .. (d) Using CLT. VnX-X) a 14 N(0,1) then X 4N y) - 9 = 0, 1, 2, ... (e) P(Y = yly)= The pdf of vis f() = 8e-9; 7 > 0 = 0 otherwise E(Y) = E(E(Y )) = E(Var(Y|)) = E(7) = theta Var(E(Y|Y)) = Var() = 1+A Var(Y) = E(Var(Y )) +Var(E(Y )) =