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Unformatted text preview: 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 counterexample. 2. Consider three independent Poisson arrival processes N i ( t ) ,t 0 with rates 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 { , 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 ....
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This note was uploaded on 10/02/2009 for the course STAT 87528 taught by Professor Pitman,jim during the Spring '09 term at University of California, Berkeley.
 Spring '09
 Pitman,Jim
 Statistics

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