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# ex_10 - Given 2 distributions F 1 and F 2 and the matrix P...

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Spring 2009 OR3510/5510 Final Problem Set 10; due April 27 as usual. Reading: We are transitioning out of Chapter 7, have browsed in Chapter 8 and will ﬁnish with some material on Brownian motion and possibly ﬁnance. x/y=page x, problem y in Ross. (1) 484/21 (2) 485/26 (3) 486/34 (4) 558/1 (Part (a) should look familiar.) (5) 558/3 (6) 559/6 (7) Re length biased sampling: Let X be uniform on [0 , 1]. Then [0 , 1] = [0 ,X ] ( X, 1] . By symmetry, each subinterval has mean length 1 / 2. Now pick one of the subintervals at random in the following way: Let Y be independent of X and uniformly distributed on [0 , 1], and pick the subinterval [0 ,X ] or ( X, 1] that Y falls in. Let L be the length of the interval so chosen so that L = ( X, if Y X 1 - X, if Y > X. Find EL . (8) Two–state semi–Markov process:
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Unformatted text preview: Given 2 distributions F 1 and F 2 and the matrix P = α 1-α 1-β β < α < 1 , < β < 1. A stochastic process { X ( t ) ,t ≥ } has state space { 1 , 2 } and moves according to the following scheme: When the process enters i , it stays in i a random amount of time governed by distribution F i ( x ) , 1 = 1 , 2 . Then given that this sojourn time is at an end, the process jumps to state j (possibly j = i ) with probability p ij which is the ( i,j )–th entry of the matrix P . Sojourn times are conditionally independent given you know the state the process is sojourning in. Use Smith‘s theorem to calculate lim t →∞ P [ X ( t ) = j ] for j = 1 , 2. 1...
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