Example 110 two stage markov chains in a markov chain

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Unformatted text preview: arkov chains. In a Markov chain the distribution of Xn+1 only depends on Xn . This can easily be generalized to case in which the distribution of Xn+1 only depends on (Xn , Xn 1 ). For a concrete example consider a basketball player who makes a shot with the following probabilities: 1/2 if he has missed the last two times 2/3 if he has hit one of his last two shots 3/4 if he has hit both of his last two shots To formulate a Markov chain to model his shooting, we let the states of the process be the outcomes of his last two shots: {HH, HM, M H, M M } where M is short for miss and H for hit. The transition probability is HH HM MH MM HH 3/4 0 2/ 3 0 HM 1/4 0 1/3 0 MH 0 2 /3 0 1 /2 MM 0 1/3 0 1/2 To explain suppose the state is HM , i.e., Xn 1 = H and Xn = M . In this case the next outcome will be H with probability 2/3. When this occurs the next state will be (Xn , Xn+1 ) = (M, H ) with probability 2/3. If he misses an event of probability 1/3, (Xn , Xn+1 ) = (M, M ). The Hot Hand is a phenomenon k...
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