4. a) The random variable X is exponentially distributed with mean B. Let Y be the [20 Marks] integer part of X , that is, if X e [n, n + 1), then Y = n. (i) Find fy(y), the probability function of Y. (ii) Derive Gy(2), the probability generating function of Y. (iii) Hence find E (Y) and E (Y?). (iv) Show that var(Y) b) For any real number u, define ut = max{u, 0} by u+ u if u 2 0 10 if u