Elements of the Theory of Markov Processes and Their ApplicationsGraduate-level text and reference in probability, with numerous scientific applications. Nonmeasure-theoretic introduction to theory of Markov processes and to mathematical models based on the theory. Appendixes. Bibliographies. 1960 edition. |
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Page 73
... o ( At ) . ( 3 ) The probability of no change in ( t , t + At ) is 1 - λ At + o ( At ) . The above probabilities are independent of the state of the system . Let P2 ( t ) = P { X ( t ) = x } x = 0 , 1 , 2 , ... In view of the above ...
... o ( At ) . ( 3 ) The probability of no change in ( t , t + At ) is 1 - λ At + o ( At ) . The above probabilities are independent of the state of the system . Let P2 ( t ) = P { X ( t ) = x } x = 0 , 1 , 2 , ... In view of the above ...
Page 182
... o ( At ) . I 2. The probability of a unit decrease in the population size of S1 in the interval ( t , t + At ) , given that there are x individuals in S1 at time t , is μ At + o ( At ) . Мх 2 3. The probability of a unit increase in the ...
... o ( At ) . I 2. The probability of a unit decrease in the population size of S1 in the interval ( t , t + At ) , given that there are x individuals in S1 at time t , is μ At + o ( At ) . Мх 2 3. The probability of a unit increase in the ...
Page 367
... o ( At ) ( 2 ) P { ( X1 , X2 , X3 ) → ( X1 − 1 , x2 + 1 , x3 + 1 ) } = - = k2 ( r — x3 ) x1 △ t + o ( At ) www - ( 8.31 ) ( 3 ) P { ( X1 , X2 , X3 ) → ( x1 + 1 , x2 − 1 , X3 ) } = k2 ( 8 — X1 — X2 ) x2 At + o ( At ) X1 The ...
... o ( At ) ( 2 ) P { ( X1 , X2 , X3 ) → ( X1 − 1 , x2 + 1 , x3 + 1 ) } = - = k2 ( r — x3 ) x1 △ t + o ( At ) www - ( 8.31 ) ( 3 ) P { ( X1 , X2 , X3 ) → ( x1 + 1 , x2 − 1 , X3 ) } = k2 ( 8 — X1 — X2 ) x2 At + o ( At ) X1 The ...
Contents
Introduction | 1 |
Processes Discrete in Space and Time | 9 |
Processes Discrete in Space and Continuous in Time | 57 |
Copyright | |
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Other editions - View all
Elements of the Theory of Markov Processes and Their Applications A. T. Bharucha-Reid Limited preview - 2012 |
Elements of the Theory of Markov Processes and Their Applications Albert T. Bharucha-Reid Limited preview - 1997 |
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absorber applications assume assumptions asymptotic birth process birth-and-death process boundary branching processes cascade process cascade theory coefficient collision consider counter defined denote the number denote the probability deterministic differential equation diffusion equations diffusion processes discrete branching process distribution function E₁ electron-photon cascades energy epidemic exists expression Feller finite functional equation given Hence initial condition integral equation interval 0,t ionization Kolmogorov equations Laplace transform Let the random machine Markov chain Markov processes Math mathematical matrix Mellin transform method Monte Carlo methods neutron nonnegative nucleon number of individuals o(At obtain P₁ photon Poisson process population probability distribution problem Proc product density queueing system r₁ radiation Ramakrishnan random variable random variable X(t random walk recurrent satisfies sequence Statist stochastic model Stochastic Processes t₁ t₂ Takács tion transition probabilities X₁ zero дх