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Unformatted text preview: E3106, Solutions to Homework 3 Columbia University Problem 4.18 . De f ne X n = if coin 1 is F ipped on the nth day 1 if coin 2 is F ipped on the nth day then { X n ,n } is an irreducible ergodic Markov chain with transition proba bility matrix P = . 6 . 4 . 5 . 5 . The limiting probabilities satisfy + 1 = 1 = 0 . 6 + 0 . 5 1 These solve to yield = 5 9 , 1 = 4 9 . (a) The desired proportion is equal to = 5 9 . (b) P 4 = . 6 . 4 . 5 . 5 4 = . 5556 . 4444 . 5555 . 4445 . The desired probability is equal to P 4 01 = 0 . 4444. Problem 4.20 We have an irreducible and aperiodic Markov chain with a f nite number of states { ,...,M } such that M X i =0 P ij = 1 : for all j { ,...,M } Since it has only one class with f nite number of states, the Markov chain is recurrent (remark (ii) page 193). Thus, it also positive recurrent (see page 200) as it has only f nite number of states. Hence, the limiting probabilities exist and are unqiue. Therefore, we only need to show that as i = 1 M + 1 , : for all j { ,...,M } 1 solves (4.7) on page 201. This is true as i = 1 M + 1 = 1 M + 1 M X i =0 P ij = M X i =0 1 M + 1 P ij = M X i =0 i P ij M X j =0 i = M X i =0 1 M + 1 = 1 M + 1 M X i =0 1 = M + 1 M + 1 = 1 ....
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 Spring '09

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