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02faex1 - 632 Introduction to Stochastic Processes Fall...

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632 Introduction to Stochastic Processes Fall 2002 Midterm Exam I Instructions: Hand in problem no. 1 for 60 points, and one other prob- lem for 40 points. Show calculations and justify non-obvious statements for full credit. General notation: P x ( A ) is the probability of the event A when the chain starts in state x , P μ ( A ) the probability when the initial state is random with distribution μ . T y = min { n 1 : X n = y } is the first time after 0 that the chain visits y , or if no visit to y ever happens. ρ x,y = P x ( T y < ) is the probability that the chain visits y some time after time 0, given that it started at x . N ( y ) is the number of visits to state y , not counting a possible visit at time 0. 1. Fix a constant α (0 , 1), and consider the Markov chain on the state space S = { 1 , 2 , 3 , 4 , 5 , 6 } with transition matrix P = 0 0 1 - α α 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 - α 0 α 0 (a) Draw the arrow diagram for the Markov chain. Classify the states according to recurrence and transience. Find the periods of recurrent states.

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