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. find P(X x, Y y)