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ex_2 - Spring 2009 OR3510/5510 Problem Set 2 Due Monday Feb...

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Spring 2009 OR3510/5510 Problem Set 2 Due Monday Feb 9 at 10am. You may insert in the homework box between Rhodes and Upson or (as Bill Maher would say: new rules) give it to me in PHL 101 at the beginning of class by 10:10am. Reading: By Wednesday we will be zipping in section 4.4 of the text. x/y=page x, problem y in Ross. (1) For states of the Markov chain i, j S , if i 7→ j but j 6 7→ i , then i is transient. (2) Consider the gamblers ruin chain on states { 0 , 1 , 2 , 3 , 4 } so that if 1 i 3 , p i,i +1 = 0 . 4 , p i,i - 1 = 0 . 6 , but the endpoints are absorbing: p 00 = 1 and p 44 = 1 . (a) Compute p (3) 14 . (b) Compute p (3) 10 . (c) What is the probability that starting from state 2, there is no absorbtion for 4 moves; that is, the game has not ended after 4 moves. (You might not want to do this by hand.) (3) Which states are transient and which are recurrent for the following matrices? ( a ) P 1 = 1 2 3 4 5 . 4 . 3 . 3 0 0 0 . 5 0 . 5 0 . 5 0 . 5 0 0 0 . 5 0 . 5 0 0 . 3 0 . 3 . 4 . ( b ) P 2 = 1 2 3 4 5 6
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