This book is an excellent and solid monograph on Probability. The author, Prof. A. DasGupta has a long experience in both teaching and research in Statistics and Probability. Among others, he is the author of a previous introductory book (Fundamentals of Probability: a first course) aimed at undergraduate students. The book under review can be considered as its second part, potentially designed for graduate students and researchers. The contents of the book begin with a survey of univariate and multivariate probability together with the classical statistical constructions derived from them. The approach is formal but enriched with a varied collection of worked out examples and a long list of exercises. In my opinion, this part (one third of the total) can be still used in the undergraduate courses of some universities. After this introductory part, the manual provides an appealing collection of advanced topics, from classical models and result to more modern constructions. The style is preserved, with lots of exercises and problems together with formal proofs and definitions. The title of the book shows the intention of giving the fundamentals of other topics, in particular, the applications to machine learning. It is not a book of machine learning theory, but an excellent exposition of the probabilistic and statistical milestones needed for its development. This applied flavor does not compete with the formality of the results. With respect to this point, the author decided to avoid formal use of measure theory but this decision is defended in the introduction for the sake of clarity and simplicity. The book would have had more than its current 782 pages if it had included this theoretical approach. I think that the decision is consistent, although there are some sections where the measure theoretical questions arise in such a natural way that some additional words would have fitted in perfectly.

I am convinced that the interested reader will enjoy this work. The structure of the 20 chapters is very similar. They all begin with an enlightening introduction that motivates the subsequent definitions and theorems. The examples and final exercises complete the topic. The detailed and exhaustive bibliography at the end of each chapter includes both classic and recent publications. There are some harder sections and exercises (marked with asterisk) that can be skipped in the first reading. In addition, the author gives suggestions of courses that can be covered by selected choice of chapters. The size of the book provides enough material for different approaches and orientations.

Reviewer:

Marco CASTRILLON LOPEZ

Affiliation:

Facultad de ciencias Matematicas. Universidad Complutense de Madrid