hmwk8_11

# hmwk8_11 - ISyE 3232 H Ayhan Stochastic Manufacturing and...

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ISyE 3232 Stochastic Manufacturing and Service Systems Fall 2011 H. Ayhan Homework 8 October 24, 2011 (Due: at the start of class on Monday, October 31) 1. Consider two stocks. Stock 1 always sells for \$10 or \$20. If stock 1 is selling for \$10 today, there is a .80 chance that it will sell for \$10 for tomorrow. If it is selling for \$20 today, there is a .90 chance that it will sell for \$20 tomorrow. Stock 2 always sells for \$10 or \$25. If stock 2 sells today for \$10 there is a .90 chance that it will sell tomorrow for \$10. If it sells today for \$25, there is a .85 chance that it will sell tomorrow for \$25. Let X n denote the price of the 1st stock and Y n denote the price of the 2nd stock during the n th day. Assume that { X n : n 0 } and { Y n : n 0 } are discrete time Markov chains. (a) What is the transition matrix for { X n : n 0 } ? Is { X n : n 0 } irreducible? (b) What is the transition matrix for { Y n : n 0 } ? Is { Y n : n 0 } irreducible? (c) What is the stationary distribution of { X n : n 0 } ? (d) What is the stationary distribution of { Y n : n 0 } ? (e) On January 1st, your grand parents decide to give you a gift of 300 shares of either Stock 1 or Stock 2. You are to pick one stock. Once you pick the stock you cannot change your mind. To take advantage of a certain tax law, your grand parents dictate that one share of the chosen stock is sold on each trading day. Which stock should you pick to maximize your gift account by the end of the year? (Explain your answer.) 2. A Markov chain is said to be doubly stochastic if both the rows and columns of the transition matrix sum to 1. Assume that the state space is { 0 , 1 , . . . , m } , and that the Markov chain is doubly stochastic and irreducible. Determine the stationary distribution

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