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UC Berkeley Department of Electrical Engineering and Computer Sciences EECS 126: Probability and Random Processes Problem Set 7 Spring 2018 Issued: March 2, 2018 Due: Wednesday, March 7, 2018 1 . Two-State Chain with Linear Algebra Consider the Markov chain ( X n , n N ), shown in Figure 1 , where α, β (0 , 1). Figure 1: Markov chain for Problem 1 . (a) Find the probability transition matrix P . (b) Find two real numbers λ 1 and λ 2 such that there exists two non-zero vectors u 1 and u 2 such that Pu i = λ i u i for i = 1 , 2. Further, show that P can be written as P = U Λ U - 1 , where U and Λ are 2 × 2 matrices and Λ is a diagonal matrix. Hint: This is called the eigendecomposition of a matrix. (c) Find P n in terms of U and Λ for each n N . (d) Assume that X 0 = 0. Use the result in part (c) to compute the PMF of X n for all n N . (e) What does the fraction of time spent in state 0, n - 1 n i =1 { X i = 0 } , converge to (almost surely) as n → ∞ ? 2 . Reducible Markov Chain Consider the following Markov chain, for α, β, p, q (0 , 1). 0 1 2 3 4 5 1 - α α β 1 - β 1 / 2 1 / 2 1 / 2 1 / 2 p 1 - p q 1 - q 1
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(a) What are all of the communicating classes? (Two nodes x and y are said to belong to the same communicating class if x can reach y and y can reach x through paths of positive probability.) For each communicating class, classify it as recurrent or transient. (b) Given that we start in state 2, what is the probability that we will reach state 0 before state 5? (c) What are all of the possible stationary distributions of this chain? (Note that there is more than one.) (d) Suppose we start in the initial distribution π 0 := 0 0 γ 1 - γ 0 0 for some γ [0 , 1]. Does the distribution of the chain converge, and if so,
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