*A First Course in Probability with an Emphasis on Stochastic Modeling*

**Probability and Stochastic Modeling** not only covers all the topics found in a traditional introductory probability course, but also emphasizes stochastic modeling, including Markov chains, birth-death processes, and reliability models. Unlike most undergraduate-level probability texts, the book also focuses on increasingly important areas, such as martingales, classification of dependency structures, and risk evaluation. Numerous examples, exercises, and models using real-world data demonstrate the practical possibilities and restrictions of different approaches and help students grasp general concepts and theoretical results.

The text is suitable for majors in mathematics and statistics as well as majors in computer science, economics, finance, and physics. The author offers two explicit options to teaching the material, which is reflected in "routes" designated by special "roadside" markers. The first route contains basic, self-contained material for a one-semester course. The second provides a more complete exposition for a two-semester course or self-study.

"This book covers practically all essential topics from modern probability and stochastic processes, which in one or another way are included in various university programs. So, is there something making this book different or better than many others available in the market? The answer is yes! The author presents the material systematically and rigorously. Of primary importance, however, is the emphasis on the modeling aspects. The large number of well-chosen and carefully described practical cases is a tremendous advantage of the book. … strongly recommended as a basic source for a variety of courses."

—G. Schröder, *Zentralblatt MATH* 1268

"The author’s mode of presentation is well chosen. New concepts are introduced with illuminating discussion and illustrations. … The book is physically nicely presented. … this text has something new to offer, mainly for business-oriented students, and I recommend that instructors give it serious consideration when choosing a text for their courses.

—Anthony G. Pakes, *Mathematical Reviews*, September 2013

"This is a superbly written text on probability and stochastic processes for students who have had two semesters of calculus and an introductory course in linear algebra. This includes upper division students in science and engineering including statistics and mathematics, as well as students in fields such as economics and finance. In addition, it will be a wonderful book for self study for many others. Important and well-chosen examples illustrate the theory throughout, and a large body of exercises supplements the text. It gives a lucid presentation of basic probability theory, including Markov chains and martingales. A special feature of this book is a marvelous exposition of many interesting aspects of financial mathematics that are generally considered rather intricate and inaccessible at this level. This book carries the imprint of a distinguished mathematician and teacher with expertise in probability theory and many of its special applications to mathematical economics and finance. It is an outstanding addition to the field requiring only a modest background in mathematics."

—Rabi Bhattacharya, Department of Mathematics, University of Arizona, Tucson, USA

"Written in a lively and stimulating manner, the book makes a very good impression. The author, having extensive teaching experience and an undoubted literary talent, has managed to create an original introduction to modern probability theory. The successful combination of a variety of examples, exercises and applications with deep and nontrivial ideas makes the book interesting not only for beginning students, but also for professionals working with probabilistic problems. I believe that the book can serve as an ideal textbook for anyone interested in probability theory and its applications. The book will take a worthy place in the literature on probabilistic issues."

—Youri Davydov, Laboratoire Paul Painlevé, Université des Sciences et Technologies de Lille, France

"The author has produced a comprehensive introduction to probability theory and stochastic processes, including martingales and Brownian motion. The text is suitable for students with a standard background in calculus and linear algebra. The approach is rigorous without being pedantic, and the text is liberally sprinkled with examples. Throughout, there is a welcome emphasis on stochastic modeling. Of note is the fairly early introduction and use of conditional expectations.

The main text is complemented by a large collection of exercises with a wide range of difficulty. The book is in fact two-in-one, as a series of ‘roadside markers’ guides the reader through two possible courses of study, one consisting of material suitable for a one-semester course, and the other a more in-depth journey suitable for a two-semester course. This book is a welcome and attractive addition to the list of textbooks available for an upper division probability course and would even be suitable for a graduate-level introduction to non-measure-theoretic probability and stochastic processes."

—Patrick J. Fitzsimmons, Department of Mathematics, University of California-San Diego, La Jolla, USA

**Basic Notions**

Sample Space and Events

Probabilities

Counting Techniques

**Independence and Conditional Probability**

Independence

Conditioning

The Borel-Cantelli Theorem

Discrete Random Variables

Random Variables and Vectors

Expected Value

Variance and Other Moments. Inequalities for Deviations

Some Basic Distributions

Convergence of Random Variables. The Law of Large Numbers

Conditional Expectation

**Generating Functions. Branching Processes. Random Walk Revisited**

Branching Processes

Generating Functions

Branching Processes Revisited

More on Random Walk

**Markov Chains**

Definitions and Examples. Probability Distributions of Markov Chains

The First Step Analysis. Passage Times

Variables Defined on a Markov Chain

Ergodicity and Stationary Distributions

A Classification of States and Ergodicity

**Continuous Random Variables**

Continuous Distributions

Some Basic Distributions

Continuous Multivariate Distributions

Sums of Independent Random Variables

Conditional Distributions and Expectations

**Distributions in the General Case. Simulation**

Distribution Functions

Expected Values

On Convergence in Distribution and Probability

Simulation

Histograms

**Moment Generating Functions**

Definitions and Properties

Some Examples of Applications

Exponential or Bernstein-Chernoff’s Bounds

**The Central Limit Theorem for Independent Random Variables**

The Central Limit Theorem (CLT) for Independent and Identically Distributed Random Variables

The CLT for Independent Variables in the General Case

**Covariance Analysis. The Multivariate Normal Distribution. The Multivariate Central Limit Theorem**

Covariance and Correlation

Covariance Matrices and Some Applications

The Multivariate Normal Distribution

Maxima and Minima of Random Variables. Elements of Reliability Theory. Hazard Rate and Survival Probabilities

Maxima and Minima of Random Variables. Reliability Characteristics

Limit Theorems for Maxima and Minima

Hazard Rate. Survival Probabilities

**Stochastic Processes: Preliminaries**

A General Definition

Processes with Independent Increments

Brownian Motion

Markov Processes

A Representation and Simulation of Markov Processes in Discrete Time

Counting and Queuing Processes. Birth and Death Processes: A General Scheme

Poisson Processes

Birth and Death Processes

Elements of Renewal Theory

Preliminaries

Limit Theorems

Some Proofs

**Martingales in Discrete Time**

Definitions and Properties

Optional Time and Some Applications

Martingales and a Financial Market Model

Limit Theorems for Martingales

Brownian Motion and Martingales in Continuous Time

Brownian Motion and Its Generalizations

Martingales in Continuous Time

More on Dependency Structures

Arrangement Structures and the Corresponding Dependencies

Measures of Dependency

Limit Theorems for Dependent Random Variables

Symmetric Distributions. De Finetti’s Theorem

**Comparison of Random Variables. Risk Evaluation**

Some Particular Criteria

Expected Utility

Generalizations of the EUM Criterion

Appendix

References

Answers to Exercises

Index

Exercises appear at the end of each chapter.

- MAT029010
- MATHEMATICS / Probability & Statistics / Bayesian Analysis
- TEC029000
- TECHNOLOGY & ENGINEERING / Operations Research