Several years ago Yuri Suhov and Mark Kelbert published a book entitled Probability and Statistics By Example, the bulk of which consisted of worked problems from the Tripos examinations given at Cambridge University. There was some exposition of the material, but most of the pages were dedicated to simply stating problems and giving solutions. I reviewed this book for MAA Reviews and wrote at the time that "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." Indeed, I am teaching Probability this semester and I have the book on my desk and I refer to it often. So it came with pleasure when our friendly MAA Reviews editor asked if I would be interested in taking a look at the second volume.
The second volume is subtitled "Markov Chains: A primer in random processes and their applications", and the title sums up the content of the book. The early parts of the book rely heavier on pure exposition than in the first volume, but the bulk of the book again consists of worked examples and problems from actual Tripos examinations. The problems cover a wide variety of topics related to Markov processes, and there is significant time dedicated to both the discrete and the continuous situation. The authors also look at statistics related to Markov chains, including likelihood functions, Bayesian analyses, and hidden Markov models. I enjoyed reading this book a great deal, and while the book is not the best source for someone's first introduction to the topics at hand, it is a great way to flesh out the general theory with concrete examples and applications.
Darren Glass is an Assistant Professor of Mathematics at Gettysburg College whose research interests include Number Theory, Algebraic Geometry, and Cryptography. He can be reached at firstname.lastname@example.org.
Preface; Introduction: Andrei Markov and his time; 1. Discrete-time Markov chains; 2. Continuous-time Markov chains: basic theory; 3. Statistics of discrete-time Markov chains; Afterword: Pearson, Maxwell and other famous Cambridge Wranglers of the past: some lessons to be learned; Bibliography; Appendix; Index.