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 Massachusetts Institute of Technology
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 6 041  Probabilistic Systems Analysis and Applied Probability
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Massachusetts Institute of Technology Department of Electrical Engineering Computer Science 6 041 6 431 Probabilistic Systems Analysis Fall 2008 Problem Set 9 Due November 19 2008 Topics Markov Chains Text sections 7 1 7 4 1 Consider the discrete state discrete trial Markov process shown below In this problem we will call transitions 1 2 2 3 3 1 clockwise transitions Similarly we will classify transitions 3 2 2 1 1 3 as counterclockwise transitions We start observing the process after a very large number of trials after it begins a Either calculate the steady state probabilities or explain why they do not exist b What is the probability there is a clockwise transition on the first trial we observe c What is the probability there is a counterclockwise transition on the first trial we observe d What is the probability there is a clockwise transition on the first change of state we observe e What is the probability there is a counterclockwise transition on the first change of state we observe f What is the conditional probability the process was in state 1 when we began observing the system given a clockwise transition occurred on the first trial we observed g What is the conditional probability the process was in state 1 when we began observing the system given a clockwise transition occurred on the first change of state we observed 2 The outcomes of successive flips of a particular coin are dependent and are found to be described fully by the conditional probabilities P Hn 1 Hn 3 4 P Tn 1 Tn 2 3 Page 1 of 3 Massachusetts Institute of Technology Department of Electrical Engineering Computer Science 6 041 6 431 Probabilistic Systems Analysis Fall 2008 where we have used the notation Event Hk Heads on kth toss Event Tk Tails on kth toss We know that the first toss came up heads a Determine the probability that the first tail will occur on the kth toss k 2 3 4 For parts b to d assume 5000 is a sufficiently large number for the probability of being in any particular state
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