Question

Problem 2: 15 points (= 5 + 5 + 5] Assume that an integer-valued random variable N has distribution P[N = n] = - valid for n > 0. Once N = n is observed, a variable X is defined as the number of successes after M = (2N+4) independent binary trials with success rate q = 0.6, or equivalently, for integer k, such that 0 Sk <2N + 4, the conditional distribution of X is: (2N+4)! P ( X = k N = n] = 1 1. Find expected value of X, that is E[X] 2. Derive variance of X, that is: Var [X] 3. Determine the marginal second moment of X, that is EX) k!. (2N +4 1-(0.6)* . (0.4)2N+4-6