Probability and Random Processes

This third edition of this successful text gives a rigorous and extensive introduction to probability theory and an account in some depth of the most important random processes. It includes various topics which are suitable for undergraduate courses, but are not routinely taught.It is suitable for students of probability at all levels. There are four main aims: 1) to provide a thorough but straightforward account of basic probability, giving the reader a natural feel for the subject unburdened by oppressive technicalities, 2) to discuss important random processes in depth with many examples. 3) to cover a range of important but less routine topics, 4) to impart to the beginner the flavour of more advanced work.The book begins with basic ideas common to many undergraduate courses in mathematics, statistics and the sciences; it concludes with topics usually found at graduate level. The ordering and numbering of material in this third edition has been mostly preserved from the second. Minor alterations and additions have been added for clearer exposition. Highlights include new sections on sampling and Markov chain Monte Carlo, geometric probability, coupling and Poisson approximation, large deviations, spatial Poisson processes, renewal-reward, queuing networks, stochastic calculus, Ito's formula and option pricing in the Black- Scholes model for financial markets.In addition there are many (nearly 400) new exercises and problems that are entertaining and instructive; their solutions can be found in the companion volume 'One Thousand Exercises in Probability', (OUP). Title page 1 Events and their probabilities 1.1 Introduction 1.2 Events as sets 1.3 Probability 1.4 Conditional probability 1.5 Independence 1.6 Completeness and product spaces 1.7 Worked examples 1.8 Problems 2 Random variables and their distributions 2.1 Random variables 2.2 The law of averages 2.3 Discrete and continuous variables 2.4 Worked examples 2.5 Random vectors 2.6 Monte Carlo simulation 2.7 Problems 3 Discrete random variables 3.1 Probability mass functions 3.2 Independence 3.3 Expectation 3.4 Indicators and matching 3.5 Examples of discrete variables 3.6 Dependence 3.7 Conditional distributions and conditional expectation 3.8 Sums of random variables 3.9 Simple random walk 3.10 Random walk: counting sample paths 3.11 Problems 4 Continuous random variables 4.1 Probability density functions 4.2 Independence 4.3 Expectation 4.4 Examples of continuous variables 4.5 Dependence 4.6 Conditional distributions and conditional expectation 4.7 Functions of random variables 4.8 Sums of random variables 4.9 Multivariate normal distribution 4.10 Distributions arising from the normal distribution 4.11 Sampling from a distribution 4.12 Coupling and Poisson approximation 4.13 Geometrical probability 4.14 Problems 5 Generating functions and their applications 5.1 Generating functions 5.2 Some applications 5.3 Random walk 5.4 Branching processes 5.5 Age-dependent branching processes 5.6 Expectation revisited 5.7 Characteristic functions 5.8 Examples of characteristic functions 5.9 Inversion and continuity theorems 5.10 Two limit theorems 5.11 Large deviations 5.12 Problems 6 Markov chains 6.1 Markov processes 6.2 Classification of states 6.3 Classification of chains 6.4 Stationary distributions and the limit theorem 6.5 Reversibility 6.6 Chains with finitely many states 6.7 Branching processes revisited 6.8 Birth processes and the Poisson process 6.9 Continuous-time Markov chains 6.10 Uniform semigroups 6.11 Birth-death processes and imbedding 6.12 Special processes 6.13 Spatial Poisson processes 6.14 Markov chain Monte Carlo 6.15 Problems 7 Convergence of random variables 7.1 Introduction 7.2 Modes of convergence 7.3 Some ancillary results 7.4 Laws of large numbers 7.5 The strong law 7.6 The law of the iterated logarithm 7.7 Martingales 7.8 Martingale convergence theorem 7.9 Prediction and conditional expectation 7.10 Uniform integrability 7.11 Problems 8 Random processes 8.1 Introduction 8.2 Stationary processes 8.3 Renewal processes 8.4 Queues 8.5 The Wiener process 8.6 Existence of processes 8.7 Problems 9 Stationary processes 9.1 Introduction 9.2 Linear prediction 9.3 Autocovariances and spectra 9.4 Stochastic integration and the spectral representation 9.5 The ergodic theorem 9.6 Gaussian processes 9.7 Problems 10 Renewals 10.1 The renewal equation 10.2 Limit theorems 10.3 Excess life 10.4 Applications 10.5 Renewal-reward processes 10.6 Problems 11 Queues 11.1 Single-server queues 11.2 M/M/1 11.3 M/G/1 11.4 G/M/1 11.5 G/G/1 11.6 Heavy traffic 11.7 Networks of queues 11.8 Problems 12 Martingales 12.1 Introduction 12.2 Martingale differences and Hoeffding's inequality 12.3 Crossings and convergence 12.4 Stopping times 12.5 Optional stopping 12.6 The maximal inequality 12.7 Backward martingales and continuous-time martingales 12.8 Some examples 12.9 Problems 13 Diffusion processes 13.1 Introduction 13.2 Brownian motion 13.3 Diffusion processes 13.4 First passage times 13.5 Barriers 13.6 Excursions and the Brownian bridge 13.7 Stochastic calculus 13.8 The Itô integral 13.9 Itô's formula 13.10 Option pricing 13.11 Passage probabilities and potentials 13.12 Problems Appendix I. Foundations and notation Appendix II. Further reading Appendix III. History and varieties of probability Appendix IV. John Arbuthnot's Preface to Of the laws of chance (1692) Appendix V. Table of distributions Appendix VI. Chronology Bibliography Notation Index The third edition of this text gives a rigorous introduction to probability theory and the discussion of the most important random processes in some depth. It includes various topics which are suitable for undergraduate courses, but are not routinely taught. It is suitable to the beginner, and should provide a taste and encouragement for more advanced work.There are four main aims: 1) to provide a thorough but straightforward account of basic probability, giving the reader a natural feel for the subject unburdened by oppressive technicalities, 2) to discuss important random processes in depth with many examples. 3) to cover a range of important but less routine topics, 4) to impart to the beginner the flavour of more advanced work. The books begins with basic ideas common to many undergraduate courses in mathematics, statistics and the sciences; in concludes with topics usually found at graduate level. The ordering and numbering of material in this third edition has been mostly preserved from the second. Minor alterations and additions have been added for clearer exposition. 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.
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 sections on 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 Exercises in Probability. 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 sections on 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 Exercises in Probability . This textbook provides a wide-ranging and entertaining indroduction to probability and random processes and many of their practical applications. It includes many exercises and problems with solutions