Let V be an exponential random variable with unit rate and W be a random integer questions

15. Let V be an exponential random variable with unit rate and W be a random integer such that P(W = 0) =1 - P(W = 2) = 1/2. Assume that V and W are independent. Define two random variables, X and Y', as follows: 6 . X = (V + W)/2; . toss a fair coin; if a head turns up, let Y = V; otherwise let Y = W. (a) Show that for any r, y 2 0, (7 (3 -e-2 - 2e-" + 1(y > 2} (1 -e-2+2) , x> (y+2)/2. P(X 1}(1-e-2+2) y > max { 2x, 2). (b) Deduce from (a) the marginal edf's of X and Y', respectively. (i) Is X a continuous random variable? If so, derive its pdf. (ii) Is Y a continuous random variable? If so, derive its pdf