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# ex-4 - Spring 2011 OR3510/5510 Problem Set 4 Due Monday Feb...

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Spring 2011 OR3510/5510 Problem Set 4 Due Monday Feb 28 at noon. Reading: We will finish Branching Processes (Sec 4.7 Ross) and then do absorption probabilities for Markov chains (Sec 4.6). We will move next to the Poisson process but probably not this week. (1) For a branching process, calculate the extinction probability when p 0 = 1 / 6 , p 1 = 1 / 2 , p 3 = 1 / 3 . (2) Consider a branching process having μ < 1. Show that if X 0 = 1 , then the expected number of individuals that ever exist in this population is given by 1 / (1 - μ ). What if X 0 = n ? (3) Harry lets his health habits slip during a depressed period and discovers spots growing be- tween his toes. The spots evolve according to a branching process with generating function P ( s ) = . 15 + . 05 s + . 03 s 2 + . 07 s 3 + . 4 s 4 + . 25 s 5 + . 05 s 6 . (Thus one can read ( p 0 , . . . , p 6 ) from the coefficients of P ( s ).) Will the spots survive? With what probability? (4) Each morning an individual leaves his house and goes for a run. He is equally likely to leave either from his front or back door. Upon leaving the house, he chooses a pair of running shoes (or goes running barefoot if there are no shoes at the door from which he departed).

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ex-4 - Spring 2011 OR3510/5510 Problem Set 4 Due Monday Feb...

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