This book on probability theory and mathematical statistics is designed for a three-quarter course meeting 4 hours per week or a two-semester course meeting 3 hours per week. It is designed primarily for advanced seniors and beginning graduate students in mathematics, but it can also be used by students in physics and engineering with strong mathematical backgrounds. Let me emphasize that this is a mathematics text and not a “cookbook.” It should not be used as a text for service courses.

The mathematics prerequisites for this book are modest. It is assumed that the reader has had basic courses in set theory and linear algebra and a solid course in advanced calculus. No prior knowledge of probability and/or statistics is assumed.

My aim is to provide a solid and well-balanced introduction to probability theory and mathematical statistics. It is assumed that students who wish to do graduate work in probability theory and mathematical statistics will be taking, concurrently with this course, a measure-theoretic course in analysis if they have not already had one. These students can go on to take advanced-level courses in probability theory or mathematical statistics after completing this course.

This book consists of essentially three parts, although no such formal divisions are designated in the text. The first part consists of Chapters 1 through 6, which form the core of the probability portion of the course. The second part, Chapters 7 through 11, covers the foundations of statistical inference. The third part consists of the remaining three chapters on special topics. For course sequences that separate probability and mathematical statistics, the first part of the book can be used for a course in probability theory, followed by a course in mathematical statistics based on the second part and, possibly, one or more chapters on special topics.

The reader will find here a wealth of material. Although the topics covered are fairly conventional, the discussions and special topics included are not. Many presentations give far more depth than is usually the case in a book at this level. Some special features of the book are the following:

1. A well-referenced chapter on the preliminaries.
2. About 550 problems, over 350 worked-out examples, about 200 remarks, and about 150 references.
3. An advance warning to reader wherever the details become too involved. They can skip the later portion of the section in question on first reading without destroying the continuity in any way.
4. Many results on characterizations of distributions (Chapter 5).
5. Proof of the central limit theorem by the method of operators and proof of the strong law of large numbers (Chapter 6).
6. A section on minimal sufficient statistics (Chapter 8).
7. A chapter on special tests (Chapter 10).
8. A careful presentation of the theory of confidence intervals, including Bayesian intervals and shortest-length confidence intervals (Chapter 11).
9. A chapter on the general linear hypothesis, which carries linear models through to their use in basic analysis of variance (Chapter 12).

10. Sections on nonparametric estimation and robustness (Chapter 13).
11. Two sections on sequential estimation (Chapter 14).

The contents of this book were used in a 1 -year (two-semester) course that I taught three times at the Catholic University of America and once in a three-quarter course at Bowling Green State University. In the fall of 1973 my colleague, Professor Eugene Lukacs, taught the first quarter of this same course on the basis of my notes, which eventually became this book. I have always been able to cover this book (with few omissions) in a 1-year course, lecturing 3 hours a week. An hour-long problem session every week is conducted by a senior graduate student.

In a book of this size there are bound to be some misprints, errors, and ambiguities of presentation. I shall be grateful to any reader who brings these to my attention.