Basic Probability Theory

This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus. Taking statistics as its major field of application, the text opens with a review of basic concepts, advancing to surveys of random variables, the properties of expectation, conditional probability and expectation, and characteristic functions. Subsequent topics include infinite sequences of random variables, Markov chains, and an introduction to statistics. Complete solutions to some of the problems appear at the end of the book.

This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus. Taking statistics as its major field of application, the text opens with a review of basic concepts, advancing to surveys of random variables, the properties of expectation, conditional probability and expectation, and characteristic functions. Subsequent topics include infinite sequences of random variables, Markov chains, and an introduction to statistics. Complete solutions to some of the problems appear at the end of the book.

The main intended audience for this book is undergraduate students in pure and applied sciences, especially those in engineering. Chapters 2 to 4 cover the probability theory they generally need in their training. Although the treatment of the subject is surely su?cient for non-mathematicians, I intentionally avoided getting too much into detail. For instance, topics such as mixed type random variables and the Dirac delta function are only brie?y mentioned. Courses on probability theory are often considered di?cult. However, after having taught this subject for many years, I have come to the conclusion that one of the biggest problems that the students face when they try to learn probability theory, particularly nowadays, is their de?ciencies in basic di?erential and integral calculus. Integration by parts, for example, is often already forgotten by the students when they take a course on probability. For this reason, I have decided to write a chapter reviewing the basic elements of di?erential calculus. Even though this chapter might not be covered in class, the students can refer to it when needed. In this chapter, an e?ort was made to give the readers a good idea of the use in probability theory of the concepts they should already know. Chapter 2 presents the main results of what is known as elementary probability, including Bayes’ rule and elements of combinatorial analysis.

This text develops the necessary background in probability theory underlying diverse treatments of stochastic processes and their wide-ranging applications. In this second edition, the text has been reorganized for didactic purposes, new exercises have been added and basic theory has been expanded. General Markov dependent sequences and their convergence to equilibrium is the subject of an entirely new chapter. The introduction of conditional expectation and conditional probability very early in the text maintains the pedagogic innovation of the first edition; conditional expectation is illustrated in detail in the context of an expanded treatment of martingales, the Markov property, and the strong Markov property. Weak convergence of probabilities on metric spaces and Brownian motion are two topics to highlight. A selection of large deviation and/or concentration inequalities ranging from those of Chebyshev, Cramer–Chernoff, Bahadur–Rao, to Hoeffding have been added, with illustrative comparisons of their use in practice. This also includes a treatment of the Berry–Esseen error estimate in the central limit theorem. The authors assume mathematical maturity at a graduate level; otherwise the book is suitable for students with varying levels of background in analysis and measure theory. For the reader who needs refreshers, theorems from analysis and measure theory used in the main text are provided in comprehensive appendices, along with their proofs, for ease of reference. Rabi Bhattacharya is Professor of Mathematics at the University of Arizona. Edward Waymire is Professor of Mathematics at Oregon State University. Both authors have co-authored numerous books, including a series of four upcoming graduate textbooks in stochastic processes with applications.

This book provides a clear and straightforward introduction to applications of probability theory with examples given in the biological sciences and engineering. The first chapter contains a summary of basic probability theory. Chapters two to five deal with random variables and their applications. Topics covered include geometric probability, estimation of animal and plant populations, reliability theory and computer simulation. Chapter six contains a lucid account of the convergence of sequences of random variables, with emphasis on the central limit theorem and the weak law of numbers. The next four chapters introduce random processes, including random walks and Markov chains illustrated by examples in population genetics and population growth. This edition also includes two chapters which introduce, in a manifestly readable fashion, the topic of stochastic differential equations and their applications.

The book is an introduction to modern probability theory written by one of the famous experts in this area. Readers will learn about the basic concepts of probability and its applications, preparing them for more advanced and specialized works.

This book provides an elementary introduction to probability theory and its applications. The emphasis is on essential probabilistic reasoning, amply motivated, explained and illustrated with a large number of carefully selected samples. The fourth edition adds material related to mathematical finance, as well as expansions on stable laws and martingales.

This compact volume equips the reader with all the facts and principles essential to a fundamental understanding of the theory of probability. It is an introduction, no more: throughout the book the authors discuss the theory of probability for situations having only a finite number of possibilities, and the mathematics employed is held to the elementary level. But within its purposely restricted range it is extremely thorough, well organized, and absolutely authoritative. It is the only English translation of the latest revised Russian edition; and it is the only current translation on the market that has been checked and approved by Gnedenko himself. After explaining in simple terms the meaning of the concept of probability and the means by which an event is declared to be in practice, impossible, the authors take up the processes involved in the calculation of probabilities. They survey the rules for addition and multiplication of probabilities, the concept of conditional probability, the formula for total probability, Bayes's formula, Bernoulli's scheme and theorem, the concepts of random variables, insufficiency of the mean value for the characterization of a random variable, methods of measuring the variance of a random variable, theorems on the standard deviation, the Chebyshev inequality, normal laws of distribution, distribution curves, properties of normal distribution curves, and related topics. The book is unique in that, while there are several high school and college textbooks available on this subject, there is no other popular treatment for the layman that contains quite the same material presented with the same degree of clarity and authenticity. Anyone who desires a fundamental grasp of this increasingly important subject cannot do better than to start with this book. New preface for Dover edition by B. V. Gnedenko.