Stochastic Processes
A nonmeasure theoretic introduction to stochastic processes. Considers its diverse range of applications and provides readers with probabilistic intuition and insight in thinking about problems. This revised edition contains additional material on compound Poisson random variables including an identity which can be used to efficiently compute moments; a new chapter on Poisson approximations; and coverage of the mean time spent in transient states as well as examples relating to the Gibb's sampler, the Metropolis algorithm and mean cover time in star graphs. Numerous exercises and problems have been added throughout the text.
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Contents
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PRELIMINARIES
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1 |
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Laplace Transforms
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15 |
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Hazard Rate Functions
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35 |
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Copyright
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approximation Azuma's inequality Brownian motion compute conditional distribution Consider continuous-time Markov chain converge counting process cycle define denote the number density distributed with mean distribution F EXAMPLE expected number exponential with rate exponentially distributed finite flips given Hence identically distributed independent and identically inequality interarrival distribution interval large numbers law of large Lemma Let X(t Let X1 limiting probabilities martingale moment generating function n₁ nonnegative number of customers number of events number of transitions obtain occurs P{X₁ P₁ P₂ Poisson distributed Poisson process Poisson random variable process N(t process with rate Proposition prove queue random walk recurrent S₁ semi-Markov process sequence server stationary probabilities Statistical stochastic process Suppose T₁ T₂ Taking expectations transition probabilities U₁ variable with mean visited Wald's equation X₁ X₂ Y₁ yields Z₁ ΣΣ