## 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 170

2 we have &{X(t)} = e−" (4.4) t 2*{X(t)} = e^* | [A(r) + u(r)]e” dr (4.5) 0 Hence, for

given A(t) and u(t) the above yield explicit

of X(t). In the special case when A(t) and u(t) are independent of time we obtain ...

2 we have &{X(t)} = e−" (4.4) t 2*{X(t)} = e^* | [A(r) + u(r)]e” dr (4.5) 0 Hence, for

given A(t) and u(t) the above yield explicit

**expressions**for the mean and varianceof X(t). In the special case when A(t) and u(t) are independent of time we obtain ...

Page 203

Hence, F(s,t) is of the form F(s,t) = 1 +} {q(rs) + h(r,s)}e^*-*** (4.1.18) a = 1 From (

4.118) a formal

, m(t) = & (X(t)} = o: = 1 as n/2 ...] o | —r(m-r-ţ-1)t - t – - act ra 4.1.19 X| 0s -" 0s s ...

Hence, F(s,t) is of the form F(s,t) = 1 +} {q(rs) + h(r,s)}e^*-*** (4.1.18) a = 1 From (

4.118) a formal

**expression**for the mean can be obtained by differentiation; that is, m(t) = & (X(t)} = o: = 1 as n/2 ...] o | —r(m-r-ţ-1)t - t – - act ra 4.1.19 X| 0s -" 0s s ...

Page 369

Needless to say, the

work with in order to obtain an explicit

moments, which provide valuable information, can be obtained in the usual way.

Needless to say, the

**expression**for the generating function is rather difficult towork with in order to obtain an explicit

**expression**for P(a,a2,..ca,t). However, themoments, which provide valuable information, can be obtained in the usual way.

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### Contents

Introduction | 1 |

Processes Discrete in Space and Continuous in Time | 57 |

Processes Continuous in Space and Time | 129 |

Copyright | |

9 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 addition applications approach arrival associated assume assumptions becomes birth boundary branching processes called cascade coefficients collision concerned condition consider constant continuous counter death defined denote density derive described determined developed differential equation diffusion discussion distribution function electron energy epidemic equal exists expected expression finite fluctuation given gives growth Hence independent individuals initial condition integral interest interval introduce Kolmogorov equations Laplace transform length limit machine Markov Markov chain Markov processes Math mathematical mean method moments necessary nucleon obtain particle particular photon Poisson population positive primary problem Proof properties queueing radiation random variable reaction refer relation represent respectively satisfies shown simple ſº solution Statist Stochastic Processes Theorem theory tion transition probabilities zero