Like many subjects in mathematics, students taking courses in probability and statistics often struggle the most when it comes to applying the theory we teach in our courses to real world problems (or quasi-real world problems involving lots of balls and urns and drawers full of socks. Therefore it seems to me that one of the best things we can do with our students is give them lots and lots of exercises and problems to try. Yuri Suhov and Mark Kelbert's new book Probability and Statistics by Example is a great source for problems, some of which are standard but many of which were new to this reviewer. They wrote the book to serve their students at Cambridge University, and the bulk of the book consists of problems from the Tripos examinations between the years 1992 and 1999. Other problems come from various sources (all well-annotated by the authors) and were written as practice problems for the Tripos exams.
The first two chapters of the book cover 'Basic Probability' which the authors define as the material that is covered in the 1A Probability Examination. This includes both discrete probability — topics ranging from Bayes' Theorem and joint distributions to the inequalities of Chebyshev, Markov and Jensen — and continuous probability — starting with uniform distributions and culminating in the Central Limit Theorem. These chapters are essentially all dealt with in problems and solutions, with brief remarks and statements of theorems sprinkled throughout. The second half of the book covers 'Basic Statistics', and takes a slightly different format. Chapter Three is about parameter estimation, and instead of being a long list of problems it is a long list of example after example on topics such as Fisher's Theorem and Bayesian Estimation. Chapter Four, dealing with hypothesis testing, is structurally even more like a typical textbook, although the authors keep their explanations concrete even when discussing somewhat abstract theorems. Chapter Five gives nearly forty problems about statistics from the Tripos examinations, and again comes with full solutions to each problem.
A word about these solutions is worth giving: the authors explain that they chose not to just give the model solutions that the Tripos examiners created but for the most part choose to instead give solutions that students have written. These solutions are sometimes not as elegant or streamlined as a model solution might be, but the authors believe — and I concur — that the solutions will make more sense and be more natural to students who read them. The book is well-written, not just for the problems but also for the remarks and historical asides that the authors choose to include. While I am not sure that it is appropriate to use as a textbook for a course, I will certainly keep it on my desk next time I teach a similar course as a good resource for problems, and I would recommend it to any student wanting supplemental material.
Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include algebraic geometry and number theory. He can be reached at firstname.lastname@example.org.
Introduction; Part A. Probability: A1. Discrete outcomes; A2. Continuous outcomes; A3. Index of problems for Part A; Part B. Statistics: B1. Parameter estimation; B2. Hypothesis testing; B3. Tripos exam questions in IB statistics (1992–2004); Tables of random variables and probability distributions; Glossary.