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Unformatted text preview: Homework 8 (Stat 620, Fall 2011) Due Thu Nov 17, in class 1. The following problems arises in molecular biology. The surface of a bacterium consists of several sites at which foreign molecules—some acceptable and some not—become attached. We consider a particular site and assume that molecules arrive at the site according to a Poisson process with parameter λ . Among these molecules a proportion α are acceptable. Unacceptable molecules stay at the site for a length of time which is exponentially distributed with parameter μ 1 , whereas an acceptable molecule remains at the site for an exponential time with departure rate μ 2 . An arriving molecule will become attached only if the site is free of other molecules. (i) What percentage of the time is the site occupied with an acceptable molecule? (ii) What fraction of arriving acceptable molecules become attached? Solution : Consider a continuous–time Markov chain with 3 states. Define states 0, 1 and 2 as the site being free, attached to an unacceptable molecule and attached to an acceptable molecule respectively. Thus, the transition rate matrix Q is Q =  λ λα λ (1 α ) μ 2 μ 2 μ 1 μ 1 ....
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 Fall '08
 Chisholm
 Graph Theory, Poisson Distribution, Probability, Poisson process, Markov chain, Continuoustime Markov process

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