Each morning Jeff leaves his house and goes for a run (to the pub). He is equall

Question

Each morning Jeff leaves his house and goes for a run (to the pub). He is equally likely to leave either from his front or back door. Upon leaving his house, he chooses a pair 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 door or back door. Assume he has 2 pairs of running shoes and let X_n be the number of pairs of running shoes at the front door before he goes for a run in the morning of the nth day, what is the state space of this process? Argue that {X_n, n greaterthanorequalto 0} is a Markov chain and determine its transition matrix.

Explanation / Answer

the state space is the number of shoes at the door he leaves through

hence state i resembles i number of shoes at door leaved through

since each state is independent of previous , it is a markov chain

the transition matrix is

( [ 3/4 , 1/4, 0] , [1/4,1/2,1/4 ] ,[0,1/4,3/4] )

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