Stochastic

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Unformatted text preview: ve to pick one of the N i balls in the other urn. The number can also decrease by 1 with probability i/N . In symbols, we have computed that the transition probability is given by p(i, i + 1) = (N i)/N, p(i, i 1) = i/N for 0 i N with p(i, j ) = 0 otherwise. When N = 4, for example, the matrix is 0 0 1 2 3 4 0 1/4 0 0 0 1 2 3 4 1 0 0 0 0 3/4 0 0 2/4 0 2 /4 0 0 3/4 0 1 /4 0 0 1 0 3 1.1. DEFINITIONS AND EXAMPLES In the first two examples we began with a verbal description and then wrote down the transition probabilities. However, one more commonly describes a Markov chain by writing down a transition probability p(i, j ) with (i) p(i, j ) 0, since they are probabilities. P (ii) j p(i, j ) = 1, since when Xn = i, Xn+1 will be in some state j . The equation in (ii) is read “sum p(i, j ) over all possible values of j .” In words the last two conditions say: the entries of the matrix are nonnegative and each ROW of the matrix sums to 1. Any matrix with properties (i) and (ii) gives rise to a Markov chain,...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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