9. (15 points) Show that if n is a positive integer then L() = n(n + 1)2 =3 k=0 10. (BONUS)[20] Consider the following discrete time process: at integer times, a marble arrives which can be one of m colors. Each color is equally likely; all events at different times are independent. As each marble arrives, it is placed into a bowl Unless there is a marble of that color already in the bowl. Then, that marble is removed from the bowl and the new marble does not go into the bowl. (The pair is placed in a separate bin for shipping). Let G, denote the number of marbles currently in the bowl after the nth marble arrives. Compute (a) the expectation E(Gn). (b) lim,+E(Gm).