Probability and random processes
This completely revised text provides a simple but rigorous introduction to probability. It discusses a wide range of random processes in some depth with many examples, and gives the beginner some flavor of more advanced work, by suitable choice of material. The book begins with basic material commonly covered in first-year undergraduate mathematics and statistics courses, and finishes with topics found in graduate courses. Important features of this edition include new and expanded sections in the early chapters, providing more illustrative examples and introducing more ideas early on; two new chapters providing more comprehensive treatment of the simpler properties of martingales and diffusion processes; and more exercises at the ends of almost all sections, with many new problems at the ends of chapters. The companion volume Probability and Random Processes: Problems and Solutions includes complete worked solutions to all exercises and problems of this edition. This proven text will be useful for mathematics and natural science undergraduates at all levels, and as a reference book for graduates and all those interested in the applications of probability theory.
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Events and their probabilities
Random variables and their distributions
Continuous random variables
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a-field arrival autocovariance function birth process birth-death process branching process calculate called characteristic function conditional expectation continuous continuous-time convergence countable deduce defined Definition denote discrete-time distributed random variables distribution function distribution with parameter ergodic event Example Exercises exists exponentially distributed function F given Hence identically distributed random independent identically distributed independent random variables inequality infinitely interarrival interval irreducible joint density function large numbers Lemma Let Xu X2 Markov chain Markov property martingale mass function non-negative non-null persistent normal distribution notation obtain particle Poisson process probability generating function probability space Problem process with intensity Proof prove queue random walk real numbers renewal process result sample paths satisfies Section sequence Show simple random walk solution stationary distribution stationary process strongly stationary submartingale subsets Suppose symmetric random walk theory transient transition matrix transition probabilities variance vector Wiener process zero means