Probability and random processes
This book gives an introduction to probability and its many practical application by providing a thorough, entertaining account of basic probability and important random processes, covering a range of important topics. Emphasis is on modelling rather than abstraction and there are new sectionson sampling and Markov chain Monte Carlo, renewal-reward, queueing networks, stochastic calculus, and option pricing in the Black-Scholes model for financial markets. In addition, there are almost 400 exercises and problems relevant to the material. Solutions can be found in One Thousand Exercisesin Probability.
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Events and their probabilities
Random variables and their distributions
Continuous random variables
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a-field arrival asserts asymptotic autocovariance function branching process calculate called Cauchy characteristic function conditional expectation constant contains continuous-time convergence countable deduce defined Definition denote distributed with parameter distribution function ergodic theorem event Example exists exponentially distributed finite function F given independent identically distributed independent variables inequality integral inter-arrival interval irreducible joint density function joint distribution Large Numbers Law of Large Lemma Markov chain Markov property martingale mass function non-negative non-null persistent normal distribution notation obtain particle Poisson distribution Poisson process probability generating function probability space Problem Proof queue length random variables random walk real numbers renewal process sample paths satisfies Section semigroup sequence solution spectral stationary distribution stationary process strongly stationary subsets suppose symmetric random walk takes values theory tion tossed transient transition probabilities variance Wiener process Xu X2 zero means