171 to see what the conditions in the last problem

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Unformatted text preview: nt, compute the transition probability. (b) 22 and 00 are absorbing states for the chain. Show that the probability of absorption in 22 is equal to the fraction of A’s in the state. (c) Let T = min{n 0 : Xn = 22 or 00} be the absorption time. Find Ex T for all states x. 1.67. Roll a fair die repeatedly and let Y1 , Y2 , . . . be the resulting numbers. Let Xn = |{Y1 , Y2 , . . . , Yn }| be the number of values we have seen in the first n rolls for n 1 and set X0 = 0. Xn is a Markov chain. (a) Find its transition probability. (b) Let T = min{n : Xn = 6} be the number of trials we need to see all 6 numbers at least once. Find ET . 1.68. Coupon collector’s problem. We are interested now in the time it takes to collect a set of N baseball cards. Let Tk be the number of cards we have to buy before we have k that are distinct. Clearly, T1 = 1. A little more thought reveals that if each time we get a card chosen at random from all N possibilities, then for k 1, Tk+1 Tk has a geometric distribution with success probability (N k )/N . Use this to show that the mean time to collect a set of N baseball P...
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