high level statistics problems in markov chains. need written solutions within 12 hours.
1. (Exercise 4.25 Revised) Each morning an individual leaves his house and goes for a run. He is equally
likely to leave either from his front or back door. Upon leaving the house, he chooses a palr of running
shoes (or goes running barefoot if there are no shoes at the door from which he departed). On his return
he is equally likely to enter, and leave his running shoes, either by the front or back door. Suppose he
owns a total of k pairs of running shoes. Let Xn be the number of pairs of runnlng shoes in the front door
before he leaves h1S house for running in the nth day. Is f Xn : n O) a Markov chain? If yes, explain
why it is and write down the state space and the transition matrix. If no, explain why it is not.
2. (Exercise 4.18 Revised) Coin 1 comes up heads with probability 0.6 and coin 2 with probability 0.5. A
coln is continually flipped until it comes up tails, at which time that coin is put aside and we start flipping
the other one.
(a) Let Xn H or T be the outcome of the nth toss. Is ( Xn : n 1) a Markov Chain?
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