Elements of the Theory of Markov Processes and Their ApplicationsThis graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition. |
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Page 2
... expression for the probability that at time t the population size is equal to 00. Hence, we seek P,(t) = 9{X(t) = :v}. To formulate the stochastic model, we assume (1) that, if at time t there are x > 0 bacteria in the population, the ...
... expression for the probability that at time t the population size is equal to 00. Hence, we seek P,(t) = 9{X(t) = :v}. To formulate the stochastic model, we assume (1) that, if at time t there are x > 0 bacteria in the population, the ...
Page 3
... expression for the mean population size (0.6) is the same as that for the population size (0.2) obtained from the deterministic model. In view of this correspondence, we can state that Eq. (0.1) describes the mean population size, while ...
... expression for the mean population size (0.6) is the same as that for the population size (0.2) obtained from the deterministic model. In view of this correspondence, we can state that Eq. (0.1) describes the mean population size, while ...
Page 10
... expression for the probability that the population size in the (n + 1)st generation, say, assumes a given value when the history 1 For abstract formulations of branching processes we refer to Otter [49] and Urbanik [58]. 2 Cf. Appendix ...
... expression for the probability that the population size in the (n + 1)st generation, say, assumes a given value when the history 1 For abstract formulations of branching processes we refer to Otter [49] and Urbanik [58]. 2 Cf. Appendix ...
Page 18
... expressions for the transition probabilities associated with a discrete branching process; however, the use of generating functions1 simplifies matters considerably and in many cases enables us to study certain properties of the process ...
... expressions for the transition probabilities associated with a discrete branching process; however, the use of generating functions1 simplifies matters considerably and in many cases enables us to study certain properties of the process ...
Page 21
... expression for 92{X,,}, we proceed as follows: Differentiation of F,,+1(s) = F,,[F(e)] with respect to 8 and putting s = 1 yields Fl-+1(1) = Fi.lF(1)]F'(1) = m"+1 On diiferentiating again, we obtain F'1'.+1(1) = F”(1)[1"$.(1)l2 + F'( ...
... expression for 92{X,,}, we proceed as follows: Differentiation of F,,+1(s) = F,,[F(e)] with respect to 8 and putting s = 1 yields Fl-+1(1) = Fi.lF(1)]F'(1) = m"+1 On diiferentiating again, we obtain F'1'.+1(1) = F”(1)[1"$.(1)l2 + F'( ...
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Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |
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