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# ps1 - MS&E 221 Ramesh Johari Problem Set 1 Due 5:00 PM in...

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MS&E 221 Problem Set 1 Ramesh Johari Due: January 19, 2011, 5:00 PM , in the basement of Huang Eng. Ctr. Reading. Read Sections 4.1, 4.2, 4.3, and 4.5.1 in Ross. Problem 1 (Probability review). Ross, Chapter 3, problem 45: Problem 2 (Probability review). Is it possible to find random variables X , Y , and Z such that P ( X > Y ) > 1 / 2 , P ( Y > Z ) > 1 / 2 , and P ( Z > X ) > 1 / 2 ? Problem 3. For each of the following, explain whether it is reasonable to model the given process as a Markov chain. (a) The daily closing value of the S&P 500. (b) The sequence of bids observed on a single eBay auction. (c) The times at which patients arrive to the emergency room at a hospital. Problem 4. Suppose X 0 , X 1 , X 2 , . . . is a sequence of i.i.d. Bernoulli random variables, with P ( X n = 0) = P ( X n = 1) = 1 / 2 . 1. For n 1 , define Y n = X n + X n - 1 . Is Y 1 , Y 2 , Y 3 , . . . a Markov chain? Explain your answer. 2. Suppose W 0 , W 1 , W 2 , . . . is a Markov chain (independent of the X n sequence) with state space { 0 , 1 } . Define V n to be the mod 2 sum of W n and X n : for n 0 , define V n = 0 if W n = X n , and V n = 1 if W n 6 = X n . Is V 0 , V 1 , V 2 , . . . a Markov chain? Explain your answer.

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ps1 - MS&E 221 Ramesh Johari Problem Set 1 Due 5:00 PM in...

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