Question

Problem 1: 15 points [= 5 + 5 + 5] Assume that a discrete random variable (N) is geometrically distributed, with P[N > n] = () for n = 0, 1, .... %3D Given N = n, a random variable (Y) has conditional distribution with density, fYIN (y|n) = • y'. e- for y > 0. n! 1. Determine the marginal density of Y. 2. Show that for any natural r, the moment E(Y'] is finite and evaluate it. 3. Derive variance of Y, that is Var (Y] = E (Y) – (E[Y])²