Problem Set_1.pdf - STAT 433/833 PROBLEM SET 1 1 Let cfw_Xn...

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STAT 433/833 PROBLEM SET 1 1. Let { X n } n =0 , 1 ,... be a Markov chain on state space { 0 , 1 , 2 , 3 , 4 } with transition matrix P = 0 1 / 3 2 / 3 0 0 0 0 0 1 / 2 1 / 2 0 0 0 1 / 4 3 / 4 1 0 0 0 0 1 / 2 0 0 0 1 / 2 . (a) Is this chain irreducible? Is it aperiodic? (b) Find the stationary probability vector. (c) Suppose the chain starts in state 0. What is the expected number of steps until it is in state 0 again? (d) Again, suppose X 0 = 0 . What is the expected number of steps until the chain is in state 3? (e) Again, suppose X 0 = 0 . What is the probability that the chain will enter state 4 before it enters state 2? 2. (Exercise 2.2, page 58 of the textbook) Consider the following Markov chain with state space S = { 0 , 1 , ... } . A sequence of positive numbers p 1 , p 2 , ... is given with i =1 p i = 1 . Whenever the chain reaches state 0 it chooses a new state according to the p i . Whenever the chain is at a state other than 0 it proceeds deterministically, one step at a time, toward 0. In other words, the chain has transition probability P x,x - 1 = 1 , x > 0 , P 0 ,x = p x , x > 0 . This is a recurrent chain since the chain keeps returning to 0 . Under what condition on the p x is the chain positive recurrent? In this case, what is the limiting probability distribution π ? (Hint: it may be easier to compute E ( T ) directly where T is the time of first return to 0 starting at 0 ). 3. (Exercise 2.5, page 58 of the textbook) Let { X n } n =0 , 1 ,... be the Markov chain with state space Z and the transition probability P n,n +1 = p, P n,n - 1 = 1 - p, where p > 1 / 2 . Assume X 0 = 0 . (a) Let Y = min { X 0 , X 1 , ... } . What is the distribution of Y ? (b) For positive integer k , let T k = min { n : X n = k } and let e ( k ) = E ( T k ) . Explain why e ( k ) = ke (1) . (c) Find e (1) . (Hint: (b) might be helpful.) 1
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2 STAT 433/833 PROBLEM SET 1 (d) Use (c) to give another proof that e (1) = if p = 1 / 2 . 4. (Exercise 2.7, page 59 of the textbook) Let { X n } n =0 , 1 ,... be a Markov chain with state space S = { 0 , 1 , 2 , ... } . For each of the following transition probabilities, state if the chain is positive recurrent, null recurrent, or transient. If it is positive recurrent, give the stationary probability distribution: (a) P x, 0 = 1 / ( x + 2) , P x,x +1 = ( x + 1) / ( x + 2) ; (b) P x, 0 = ( x + 1) / ( x + 2) , P x,x +1 = 1 / ( x + 2) ; (c) P x, 0 = 1 / ( x 2 + 2) , P x,x +1 = ( x 2 + 1) / ( x 2 + 2) . 5. (a) Let { X n } n =0 , 1 ,... be a DTMC. Let set A S be a set of states. Is the following statement always true? P ( X n +1 = x n +1 | X n A, X n - 1 = x n - 1 , ..., X 0 = x 0 ) = P ( X n +1 = x n +1 | X n A ) If yes, prove it; if no, give a counterexample.
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