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# Introduction to Probability with R

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## Book Description

Based on a popular course taught by the late Gian-Carlo Rota of MIT, with many new topics covered as well, **Introduction to Probability with R** presents R programs and animations to provide an intuitive yet rigorous understanding of how to model natural phenomena from a probabilistic point of view. Although the R programs are small in length, they are just as sophisticated and powerful as longer programs in other languages. This brevity makes it easy for students to become proficient in R.

This calculus-based introduction organizes the material around key themes. One of the most important themes centers on viewing probability as a way to look at the world, helping students think and reason probabilistically. The text also shows how to combine and link stochastic processes to form more complex processes that are better models of natural phenomena. In addition, it presents a unified treatment of transforms, such as Laplace, Fourier, and z; the foundations of fundamental stochastic processes using entropy and information; and an introduction to Markov chains from various viewpoints. Each chapter includes a short biographical note about a contributor to probability theory, exercises, and selected answers.

The book has an accompanying website with more information.

## Table of Contents

**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.*

## Reviews

… beginners should find the informal and nonthreatening presentation of the basic ideas very useful … A more advanced student could use the book as an extra source of intriguing mathematical examples, as could an instructor searching for interesting items to throw into a more conventional course. … a very interesting book …

—Technometrics, May 2009, Vol. 51, No. 2Generally, I was very impressed with this text. It gives a sold introduction to probability with many interesting applications. One of its strengths is its material on stochastic processes.

—Jim Albert, Bowling Green State University,The American Statistician, May 2009, Vol. 63, No. 2… a welcome addition. …The book is clearly written and very well-organized and it stems in part from a popular course at MIT taught by the late Gian-Carlo Rota, which was originally designed in conjunction with the author of this book. The book goes well beyond the MIT course in making extensive use of computation and R. … It would serve as an exemplary test for the first semester of a two-semester course on probability and statistics.

Introduction to Probability with Ris a well-organized course in probability theory. …

—Journal of Statistical Software, April 2009This 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. … 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. … Another aspect in which the book stands out among the competition is that discrete probability gets its due treatment. …

—Miklós Bóna, University of Florida,MAA Reviews, June 2008…a broad spectrum of probability and statistics topics ranging from set theory to statistics and the normal distribution to Poisson process to Markov chains. The author has covered each topic with an ample depth and with an appreciation of the problems faced by the modern world. The book contains a rich collection of exercises and problems … an excellent introduction to the open source software R is given in the book. … This book showcases interesting, classic puzzles throughout the text, and readers can also get a glimpse of the lives and achievements of important pioneers in mathematics. …

—From the Foreword, Tianhua Niu, Brigham and Women’s Hospital, Harvard Medical School, and Harvard School of Public Health, Boston, Massachusetts, USA