Unformatted text preview: Given 2 distributions F 1 and F 2 and the matrix P = α 1α 1β 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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 Summer '09
 RESNIK
 Probability, Probability theory, Stochastic process, OR3510/5510 Final Problem, distribution Fi, Twostate semiMarkov process

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