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15. Let V be an exponential random variable with unit rate and W be a random integer such tha 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 a, y 2 0, (3 -e-2x - 2e-v+ 1{y 2 2}(1 - e-21+2) , > (y+2)/2, P(X 1}(1-e-2+2)], y > max {2x, 2}. (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. (c) Calculate P(X > Y). (d) Calculate the correlation coefficient between X and Y