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). Cover......Page p0001.djvu Title page......Page filename5.djvu Preface to the Third Edition......Page filename5_0005.djvu Contents......Page filename5_0007.djvu 1.2 Events as sets......Page filename5_0011.djvu 1.3 Probability......Page filename5_0014.djvu 1.4 Conditional probability......Page filename5_0018.djvu 1.5 Independence......Page filename5_0023.djvu 1.6 Completeness and product spaces......Page filename5_0024.djvu 1.7 Worked examples......Page filename5_0026.djvu 1.8 Problems......Page filename5_0031.djvu 2.1 Random variables......Page filename5_0036.djvu 2.2 The law of averages......Page filename5_0040.djvu 2.3 Discrete and continuous variables......Page filename5_0043.djvu 2.4 Worked examples......Page filename5_0045.djvu 2.5 Random vectors......Page filename5_0048.djvu 2.6 Monte Carlo simulation......Page filename5_0051.djvu 2.7 Problems......Page filename5_0053.djvu 3.1 Probability mass functions......Page filename5_0056.djvu 3.2 Independence......Page filename5_0058.djvu 3.3 Expectation......Page filename5_0060.djvu 3.4 Indicators and matching......Page filename5_0066.djvu 3.5 Examples of discrete variables......Page filename5_0070.djvu 3.6 Dependence......Page filename5_0072.djvu 3.7 Conditional distributions and conditional expectation......Page filename5_0077.djvu 3.8 Sums of random variables......Page filename5_0080.djvu 3.9 Simple random walk......Page filename5_0081.djvu 3.10 Random walk: counting sample paths......Page filename5_0085.djvu 3.11 Problems......Page filename5_0093.djvu 4.1 Probability density functions......Page filename5_0099.djvu 4.2 Independence......Page filename5_0101.djvu 4.3 Expectation......Page filename5_0103.djvu 4.4 Examples of continuous variables......Page filename5_0105.djvu 4.5 Dependence......Page filename5_0108.djvu 4.6 Conditional distributions and conditional expectation......Page filename5_0114.djvu 4.7 Functions of random variables......Page filename5_0117.djvu 4.8 Sums of random variables......Page filename5_0123.djvu 4.9 Multivariate normal distribution......Page filename5_0125.djvu 4.10 Distributions arising from the normal distribution......Page filename5_0129.djvu 4.11 Sampling from a distribution......Page filename5_0132.djvu 4.12 Coupling and Poisson approximation......Page filename5_0137.djvu 4.13 Geometrical probability......Page filename5_0143.djvu 4.14 Problems......Page filename5_0150.djvu 5.1 Generating functions......Page filename5_0158.djvu 5.2 Some applications......Page filename5_0166.djvu 5.3 Random walk......Page filename5_0172.djvu 5.4 Branching processes......Page filename5_0181.djvu 5.5 Age-dependent branching processes......Page filename5_0185.djvu 5.6 Expectation revisited......Page filename5_0188.djvu 5.7 Characteristic functions......Page filename5_0191.djvu 5.8 Examples of characteristic functions......Page filename5_0196.djvu 5.9 Inversion and continuity theorems......Page filename5_0199.djvu 5.10 Two limit theorems......Page filename5_0203.djvu 5.11 Large deviations......Page filename5_0211.djvu 5.12 Problems......Page filename5_0216.djvu 6.1 Markov processes......Page filename5_0223.djvu 6.2 Classification of states......Page filename5_0230.djvu 6.3 Classification of chains......Page filename5_0233.djvu 6.4 Stationary distributions and the limit theorem......Page filename5_0237.djvu 6.5 Reversibility......Page filename5_0247.djvu 6.6 Chains with finitely many states......Page filename5_0250.djvu 6.7 Branching processes revisited......Page filename5_0253.djvu 6.8 Birth processes and the Poisson process......Page filename5_0256.djvu 6.9 Continuous-time Markov chains......Page filename5_0266.djvu 6.10 Uniform semigroups......Page filename5_0276.djvu 6.11 Birth-death processes and imbedding......Page filename5_0278.djvu 6.12 Special processes......Page filename5_0284.djvu 6.13 Spatial Poisson processes......Page filename5_0291.djvu 6.14 Markov chain Monte Carlo......Page filename5_0301.djvu 6.15 Problems......Page filename5_0306.djvu 7.1 Introduction......Page filename5_0315.djvu 7.2 Modes of convergence......Page filename5_0318.djvu 7.3 Some ancillary results......Page filename5_0328.djvu 7.4 Laws of large numbers......Page filename5_0335.djvu 7.5 The strong law......Page filename5_0339.djvu 7.6 The law of the iterated logarithm......Page filename5_0342.djvu 7.7 Martingales......Page filename5_0343.djvu 7.8 Martingale convergence theorem......Page filename5_0348.djvu 7.9 Prediction and conditional expectation......Page filename5_0353.djvu 7.10 Uniform integrability......Page filename5_0360.djvu 7.11 Problems......Page filename5_0364.djvu 8.1 Introduction......Page filename5_0370.djvu 8.2 Stationary processes......Page filename5_0371.djvu 8.3 Renewal processes......Page filename5_0375.djvu 8.4 Queues......Page filename5_0377.djvu 8.5 The Wiener process......Page filename5_0380.djvu 8.6 Existence of processes......Page filename5_0381.djvu 8.7 Problems......Page filename5_0383.djvu 9.1 Introduction......Page filename5_0385.djvu 9.2 Linear prediction......Page filename5_0387.djvu 9.3 Autocovariances and spectra......Page filename5_0390.djvu 9.4 Stochastic integration and the spectral representation......Page filename5_0397.djvu 9.5 The ergodic theorem......Page filename5_0403.djvu 9.6 Gaussian processes......Page filename5_0415.djvu 9.7 Problems......Page filename5_0419.djvu 10.1 The renewal equation......Page filename5_0422.djvu 10.2 Limit theorems......Page filename5_0427.djvu 10.3 Excess life......Page filename5_0431.djvu 10.4 Applications......Page filename5_0433.djvu 10.5 Renewal-reward processes......Page filename5_0441.djvu 10.6 Problems......Page filename5_0447.djvu 11.1 Single-server queues......Page filename5_0450.djvu 11.2 M/M/1......Page filename5_0452.djvu 11.3 M/G/1......Page filename5_0455.djvu 11.4 G/M/1......Page filename5_0461.djvu 11.5 G/G/1......Page filename5_0465.djvu 11.7 Networks of queues......Page filename5_0472.djvu 11.8 Problems......Page filename5_0478.djvu 12.1 Introduction......Page filename5_0481.djvu 12.2 Martingale differences and Hoeffding's inequality......Page filename5_0486.djvu 12.3 Crossings and convergence......Page filename5_0491.djvu 12.4 Stopping times......Page filename5_0497.djvu 12.5 Optional stopping......Page filename5_0501.djvu 12.6 The maximal inequality......Page filename5_0506.djvu 12.7 Backward martingales and continuous-time martingales......Page filename5_0509.djvu 12.8 Some examples......Page filename5_0513.djvu 12.9 Problems......Page filename5_0518.djvu 13.1 Introduction......Page filename5_0523.djvu 13.2 Brownian motion......Page filename5_0524.djvu 13.3 Diffusion processes......Page filename5_0526.djvu 13.4 First passage times......Page filename5_0535.djvu 13.5 Barriers......Page filename5_0540.djvu 13.6 Excursions and the Brownian bridge......Page filename5_0544.djvu 13.7 Stochastic calculus......Page filename5_0547.djvu 13.8 The Itô integral......Page filename5_0549.djvu 13.9 Itô's formula......Page filename5_0554.djvu 13.10 Option pricing......Page filename5_0557.djvu 13.11 Passage probabilities and potentials......Page filename5_0564.djvu 13.12 Problems......Page filename5_0571.djvu Appendix I. Foundations and notation......Page filename5_0574.djvu Appendix II. Further reading......Page filename5_0579.djvu Appendix III. History and varieties of probability......Page filename5_0581.djvu Appendix IV. John Arbuthnot's Preface to Of the laws of chance (1692)......Page filename5_0583.djvu Appendix V. Table of distributions......Page filename5_0586.djvu Appendix VI. Chronology......Page filename5_0588.djvu Bibliography......Page filename5_0590.djvu Notation......Page filename5_0593.djvu Index......Page filename5_0595.djvu Back Cover......Page p0608.djvu 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