## 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 13

If when n = 0 the probability of the system being in the state i is q(i), i.e., q(i) = ^{

X0 = i}, then the unconditional or absolute probability of the system being in the

state j at time n is

If when n = 0 the probability of the system being in the state i is q(i), i.e., q(i) = ^{

X0 = i}, then the unconditional or absolute probability of the system being in the

state j at time n is

**given**by ?u,(j)= I ?(»V*;' (ii2) i=0 Hence,**given**the q(i) and the ...Page 80

Therefore, the matrix of infinitesimal transition probabilities is

0 • • 0 -A A 0 0 0 •• 0 0 -2A IX 0 0 •• 0 0 0 -3A 3A 0 •□ A = Hence the Kolmogorov

equations for the birth process are

Therefore, the matrix of infinitesimal transition probabilities is

**given**by "0 0 0 0 00 • • 0 -A A 0 0 0 •• 0 0 -2A IX 0 0 •• 0 0 0 -3A 3A 0 •□ A = Hence the Kolmogorov

equations for the birth process are

**given**by dPti(t) dt = - V«(i) + WW) at (2.115) ...Page 182

The probability of a unit increase in the population size of Sx in the interval (t, t +

A<),

probability of a unit decrease in the population size of Sx in the interval (t, t + At),

...

The probability of a unit increase in the population size of Sx in the interval (t, t +

A<),

**given**that there are exactly x individuals in at time t, is Xx At + o{At). 2. Theprobability of a unit decrease in the population size of Sx in the interval (t, t + At),

...

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### Contents

Introduction | 1 |

Processes Discrete in Space and Time | 9 |

Processes Discrete in Space and Continuous in Time | 57 |

Copyright | |

10 other sections not shown

### 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 |

### Common terms and phrases

absorber Acad applications associated assume assumptions asymptotic birth process birth-and-death process branching processes cascade process cascade theory coefficient collision consider defined denote the number denote the probability derive determined deterministic differential equation diffusion equations diffusion processes distribution function electron-photon cascades epidemic exists expression Feller finite fluctuation problem functional equation given Hence initial condition integral equation interval ionization Kendall Kolmogorov equations Laplace transform Laplace-Stieltjes transform Let the random machine Markov chain Markov processes Math mathematical matrix mean and variance mean number Mellin transform Messel method Monte Carlo methods mutation neutron nonnegative nucleon nucleon cascades number of electrons number of individuals o(At obtain parameter photon Phys Poisson process probability distribution Proc Px(t queueing process queueing system radiation Ramakrishnan random variable random variable X(t reaction recurrent refer satisfies solution of Eq Statist stochastic model Stochastic Processes Theorem tion transition probabilities zero