Exercise 1 A transition matrix is doubly stochastic if each column ...

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Exercise 1 A transition matrix is doubly stochastic if each column sum is 1. Find the stationary distribution for a doubly stochastic chain with M states. Exercise 2 Trials are performed in sequence. If the last two trials were successes, then the next trial is a success with probability 0.8; otherwise the next trial is a success with probability 0.5. In the long run, what proportion of trials are successes? Exercise 3 Let {Xn} be a regular finite state Markov chain with transition matrix P and stationary distribution w. Define the process {Yn} by Yn = (Xn−1, Xn). Show that {Yn} is a Markov chain, and compute limn→∞ P(Yn = (i, j)) (1) Exercise 5 Consider the following transition probability matrix for a Markov chain on 5 states: P = 0.5 0.3 0 0 0.2 0 0.5 0 0 0.5 0 0.4 0.4 0.2 0 0.3 0 0.2 0 0.5 0.5 0.2 0 0 0.3 c). Compute the expected number of steps needed to first return to state 1, conditioned on starting in state 1. d). Compute the expected number of steps needed to first reach any of the states {1, 2, 5}, conditioned on starting in state 3. Exercise 7 [Snell and Grinstead] A city is divided into three areas 1, 2, 3. It is estimated that amounts u1, u2, u3 of pollution are emitted each day from these three areas. A fraction qij of the pollution from region i ends up the next day at region j. A fraction qi = 1 − P j qij > 0 escapes into the atmosphere. Let w (n) i be the amount of pollution in area i after n days. (a) Show that w (n) = u + uQ + · · · + uQn−1 . (b) Show that w (n) → w. (c) Show how to determine the levels of pollution u which would result in a prescribed level w. Exercise 8 [The gambler’s ruin] At each play a gambler has probability p of winning one unit and probability q = 1−p of losing one unit. Assuming that successive plays of the game are independent, what is the probability that, starting with i units, the gambler’s fortune will reach N before reaching 0? [Hint: define Pi to be the probability that the gambler’s fortune reaches N before reaching 0 conditioned on starting in state i. By conditioning on the first step derive a recursion relation between Pi , Pi+1 and Pi−1.] Page 422 18. Assume that a student going to certain four-year medical school in northern New England has, each year, a probability q of unking out, a probability r of having to repeat the year, and a probability p of moving on to the next year (in the fourth year, moving on means graduating). a) Form a transition matrix for this process taking as states F, 1, 2, 3, 4 and G where F stands for unking out and G for graduating, and the other states represent the year of study. b) For the case q = .1, r = .2 and p = .7 nd the time a beginning student can expect to be in the second year. How long should this student expect to be in medical school? c) Find the probability that this beginning student will graduate. Page 442 26. Let P be the transition matrix of an r-state ergodic chain. Prove that, if the diagonal entries pii are positive, then the chain is regular. 27. Prove that if P is the transition matrix of an ergodic chain, then (1/2)(I + P) is the transition matrix of a regular chain. Page 466 1. Consider the Markov chain with transition matrix P = 1/2 1/2 1/4 3/4 Find the fundamental matrix z for this chain. Compute the mean rst passage matrix using Z. 2. A study of the strengths of Ivy League football teams shows that if a school has a strong team one year it is equally likely to have a strong team or average team next year; if it has an average team, half the time it is average next year, and if it changes it is just as likely to become strong as weak; if it is weak it has 2/3 probability or remaining so and 1/3 of becoming average. a) A school has a strong team. On the average, how long will it be before it has another strong team? b) A school has a weak team; how long (on the average) must the alumni wait for a strong team? 16. Let P be the transition matrix of an ergodic Markov chain. Show that (I + P + . . . + P n−1 )(I − P + W) = I − P n + nW, and from this show that I + P + . . . Pn−1 n → W, as n → ∞

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