.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: 1. X = (V + W)/2; 2. toss a fair coin; if a head turns up, let Y = V; otherwise let Y = W.

(a)find P(X x, Y y)

(b) Deduce from (a) the marginal cdf'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.