Introduction to Probability with R

Introduction to Probability with R

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Features

  • Uses R programs and animations to convey important aspects of probability and to encourage experimentation
  • Covers the theorems of probability along with stochastic processes and the relationships among them
  • Deals with probabilistic reasoning in chapters on statistics and conditional probability
  • Introduces transforms via randomization, a unique approach to a very important subject
  • Explores entropy and information to demonstrate basic stochastic processes and the most commonly occurring distributions
  • Shows how Markov chains are a versatile tool for modeling natural phenomena
  • Includes many exercises and selected answers
  • Offers R programs, PowerPoint slides for presentations and lectures, and related web links on a supplementary website

Summary

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.

Editorial 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. 2

Generally, 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 R is a well-organized course in probability theory. …
Journal of Statistical Software, April 2009

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. … 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

 
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