## Probability and Random ProcessesThis 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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applied arrival assume bounded calculate called characteristic function coin collection conditional Consider constant contains continuous convergence deduce defined Definition density function depends distribution function equal equation event Example Exercises exists expectation exponential finite function F given gives heads Hence holds identically distributed independent indicator inequality integral interval joint Lemma length limit Markov chain martingale mass function matrix mean measure non-negative normal o-field obtain occurs origin parameter particle paths period persistent Poisson process positive possible probability Problem Proof properties prove queue random variables random walk respect result sample satisfies sequence Show simple solution space starting stationary distribution step stopping Suppose surely taking values theorem theory tosses transition uniformly values variance visits Wiener process write zero