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Publisher:

Chapman & Hall/CRC

Publication Date:

2008

Number of Pages:

363

Format:

Hardcover

Series:

Texts in Statistical Science 75

Price:

89.95

ISBN:

9781420065213

Category:

Textbook

[Reviewed by , on ]

Miklós Bóna

06/13/2008

This advanced undergraduate textbook is a pleasure to read and this reviewer will definitely consider it next time he teaches the subject. The programming language R is an open-source, freely downloadable software package that is used in the book to illustrate various examples. However, the book is well usable even if you do not have the time to include too much programming in your class. All programs of the book, and several others, are downloadable from the book's website.

While there are not as many exercises than in some competing textbooks, the exercises of this book are a lot of fun! They often have some historical background, they tell a story, and they are never routine. Every chapter also starts with historical background, helping the student realize that this subject was developed by actual people.

All classic topics that you would want to cover in an introductory probability class are covered. Statistics get one chapter, related to the normal distribution. Another aspect in which the book stands out among the competition is that discrete probability gets its due treatment.

When can I teach this class next?

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.

**FOREWORD**

**PREFACE**

**Sets, Events, and Probability**

The Algebra of Sets

The Bernoulli Sample Space

The Algebra of Multisets

The Concept of Probability

Properties of Probability Measures

Independent Events

The Bernoulli Process

The R Language

**Finite Processes**

The Basic Models

Counting Rules

Computing Factorials

The Second Rule of Counting

Computing Probabilities

**Discrete Random Variables**

The Bernoulli Process: Tossing a Coin

The Bernoulli Process: Random Walk

Independence and Joint Distributions

Expectations

The Inclusion-Exclusion Principle

**General Random Variables**

Order Statistics

The Concept of a General Random Variable

Joint Distribution and Joint Density

Mean, Median and Mode

The Uniform Process

Table of Probability Distributions

Scale Invariance

**Statistics and the Normal Distribution**

Variance

Bell-Shaped Curve

The Central Limit Theorem

Significance Levels

Confidence Intervals

The Law of Large Numbers

The Cauchy Distribution

**Conditional Probability**

Discrete Conditional Probability

Gaps and Runs in the Bernoulli Process

Sequential Sampling

Continuous Conditional Probability

Conditional Densities

Gaps in the Uniform Process

The Algebra of Probability Distributions

**The Poisson Process**

Continuous Waiting Times

Comparing Bernoulli with Uniform

The Poisson Sample Space

Consistency of the Poisson Process

**Randomization and Compound Processes**

Randomized Bernoulli Process

Randomized Uniform Process

Randomized Poisson Process

Laplace Transforms and Renewal Processes

Proof of the Central Limit Theorem

Randomized Sampling Processes

Prior and Posterior Distributions

Reliability Theory

Bayesian Networks

**Entropy and Information**

Discrete Entropy

The Shannon Coding Theorem

Continuous Entropy

Proofs of Shannon’s Theorems

**Markov Chains**

The Markov Property

The Ruin Problem

The Network of a Markov Chain

The Evolution of a Markov Chain

The Markov Sample Space

Invariant Distributions

Monte Carlo Markov Chains

**appendix A: Random Walks**

Fluctuations of Random Walks

The Arcsine Law of Random Walks

**Appendix B: Memorylessness and Scale-Invariance**

Memorylessness

Self-Similarity

**References**

**Index**

*Exercises and Answers appear at the end of each chapter.*

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