When trying to evaluate a textbook, there are a couple of principles that I use:
In the case of Probability, the first standard is met without too much trouble. The obligatory chapter on set theory is a bit clunky, but that’s certainly not a problem unique to this book. (Sometime, the mathematical community should commission a definitive and engaging version of this material that all authors could tap into.) As a textbook for a standard upper-division probability course, this book can hold its own against any comparable text.
In the foreword, Miller notes that his book tries to break new ground by focusing on probability as a modeling tool and by incorporating material on statistical inference throughout. In the first of these quests, he has included examples of stochastic models in the last four chapters, where the probability theory developed earlier and throughout is illustrated in a rich array of applications. The applications deliberately focus on problems not involving dice and cards — which is at once a reasonable choice and a decision that leaves considerable room for the collection of modeling problems that are introduced.
The material on statistical inference is especially well-done. There is no need with this book to make a choice between probability and statistics; though the focus is on probability, statistical topics are present and well-integrated. A probability student can come away with an understanding of the interplay between probability and statistics in such areas as hypothesis testing and queuing theory.
This book more than lives up to its ambitious title.
Mark Bollman (email@example.com) is an assistant professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted.
To the Student.
To the Instructor.
Chapter 1. Modeling.
1.1 Choice and Chance.
1.2 The Model Building Process.
1.3 Modeling in the Mathematical Sciences.
1.4 A First Look at a Probability Model: The Random Walk.
1.5 Brief Applications of Random Walks.
Chapter 2. Sets and Functions.
2.1 Operations with Sets.
2.3 The Probability Function and the Axioms of Probability.
2.4 Equally Likely Sample Spaces and Counting Rules.
Chapter 3. Probility Laws I: Building on the Axioms.
3.1 The Complement Rule.
3.2 The Addition Rule.
3.3 Extensions and Additional Results.
Chapter 4. Probility Laws II: Results of Conditioning.
4.1 Conditional Probability and the Multiplication Rule.
4.2 Independent Events.
4.3 The Theorem of Total Probabilities and Bayes' Rule.
4.4 Problems of Special Interest: Effortful Illustrations of the Probability Laws.
Chapter 5. Random Variables and Stochastic Processes.
5.1 Roles and Types of Random Variables.
5.3 Roles, Types, and Characteristics of Stochastic Processes.
Chapter 6. Discrete Random Variables and Applications in Stochastic Processes.
6.1 The Bernoulli and Binomial Models.
6.2 The Hypergeometric Model.
6.3 The Poisson Model.
6.4 The Geometric and Negative Binomial.
Chapter 7. Continuous Random Variables and Applications in Stochastic Processes.
7.1 The Continuous Uniform Model.
7.2 The Exponential Model.
7.3 The Gamma Model.
7.4 The Normal Model.
Chapter 8. Covariance and Correlation Among Random Variables.
8.1 Joint, Marginal and Conditional Distributions.
8.2 Covariance and Correlation.
8.3 Brief Examples and Illustrations in Stochastic Processes and Times Series.