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# 09fin - Statistics 150(Stochastic Processes Final Exam...

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Statistics 150 (Stochastic Processes): Final Exam, Spring 2009. J. Pitman, U.C. Berkeley. 1. Suppose that a Markov matrix P indexed by a finite set has the property that for each state i : X j 6 = i P ( i, j ) = X j 6 = i P ( j, i ) . a) What does this property imply about the stationary distribution of the Markov chain? b) Does this property imply that the stationary distribution is unique? If so, sketch a proof, and if not provide a counter-example. 2. Consider three independent Poisson arrival processes N i ( t ) , t 0 with rates λ i for i = 1 , 2 , 3, all starting at N i (0) = 0. Let T 31 be the least t 0 such that N 3 ( t ) = 1, and let X i = N i ( T 31 ) for i = 1 , 2. a) Describe the distribution of X i for each i = 1 , 2. b) Describe the joint distribution of X 1 and X 2 . c) Find E ( X 2 | X 1 ). 3. Let ( X n ) be a Markov chain with state space { 0 , 1 , . . . , 2 N } for some positive integer N with transition matrix P such that P ( i, i - 1) = P ( i, i + 1) = 1 / 2 for 1 i 2 N - 1 , P (0 , N ) = P (2 N, N ) = 1 . a) Write down the equations satisfied by the stationary distribution π for this Markov chain, with special attention to the equations associated with states 0 , N

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09fin - Statistics 150(Stochastic Processes Final Exam...

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