Hw3-ORIE3510_S12 - ORIE 3510 Homework 3 Instructor: Mark E....

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ORIE 3510 – Homework 3 Instructor: Mark E. Lewis due 2PM, Wednesday February 15, 2012 (ORIE Hallway drop box) 1. Consider the stochastic process X n = min( X n - 1 , X n - 2 ) + ǫ n with X 0 = 0, X 1 = 0 and ǫ n is a random variable taking values - 1 and 1 with probability 1 / 2, and { ǫ n , n 0 } are i.i.d. (a) Is X n a Markov chain? (b) Based on X n , construct a Markov chain. 2. Let { X n , n 0 } and { Y n , n 0 } be two independent Markov chains each with the same discrete space X and same transition matrix. Defne the process { Z n , n 0 } = { ( X n , Y n ) , n 0 } with the state space X × X . Show that { Z n , n 0 } is a Markov chain and give the transition matrix. 3. Suppose that { X n , n 0 } is a Markov chain. Show that { Y n , n 1 } = { ( X n , X n - 1 ) , n 1 } is a Markov chain and give its transition matrix. 4. Suppose that { Z n , n 1 } are i.i.d. random variables representing outcomes o± successive throws o± a ±air die. Defne X n = max { Z 1 , Z
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Hw3-ORIE3510_S12 - ORIE 3510 Homework 3 Instructor: Mark E....

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