Question

Problem 1: 15 points (= 5 + 5 + 5] You are given that an integer-valued random variable N has distribution such that P[N 2 n] = (%)' for n 20. Non-negative random variables X = {Xk: k = 1, 2, ...,} do not depend on N and share the common distribution with density f*(x) = (216) - 23.-6x valid for 2 > 0 Consider a random sum, Xk. k=1 1. Derive expectation of S, that is: ES 2N+5 2. Evaluate variance of S, that is: Var (S] 3. Find the marginal second moment of S, that is: E(S)