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: . 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 x,y 20, P(X (y+2)/2, LE [-e2+ 1-6+2+1{r > 1}(1 e~2+2)] y > max{2x, 2} 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: . 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 x,y 20, P(X (y+2)/2, LE [-e2+ 1-6+2+1{r > 1}(1 e~2+2)] y > max{2x, 2}