Essentials of Probability Theory for Statisticians provides graduate students with a rigorous treatment of probability theory, with an emphasis on results central to theoretical statistics. It presents classical probability theory motivated with illustrative examples in biostatistics, such as outlier tests, monitoring clinical trials, and using adaptive methods to make design changes based on accumulating data. The authors explain different methods of proofs and show how they are useful for establishing classic probability results.
After building a foundation in probability, the text intersperses examples that make seemingly esoteric mathematical constructs more intuitive. These examples elucidate essential elements in definitions and conditions in theorems. In addition, counterexamples further clarify nuances in meaning and expose common fallacies in logic.
This text encourages students in statistics and biostatistics to think carefully about probability. It gives them the rigorous foundation necessary to provide valid proofs and avoid paradoxes and nonsensical conclusions.
Table of Contents
Introduction. Size Matters. The Elements of Probability Theory. Random Variables and Vectors. Integration and Expectation. Modes of Convergence. Laws of Large Numbers. Central Limit Theorems. More on Convergence in Distribution. Conditional Probability and Expectation. Applications. Appendices. Index.
" The book Essentials of Probability Theory for Statisticians does not try to compete with probability textbooks like Billingsley (2012) or Chung (2001), but targets a particular audience: graduate students in statistics who need to quickly learn the essentials of probability theory to make rigorous arguments in statistics. The book does not try to give a full introduction to measure theory but instead focuses on the essentials that are needed by statisticians. . . . I think that this textbook fills an important niche: It provides a concise summary of the essentials of probability theory that are needed by statisticians and at the same time relates these concepts to important applications in statistics. Hence, the reader learns to appreciate the interplay between probability and statistics. The book is well written and uses engaging language and plenty of examples and illustrations. Overall, I enjoyed teaching from this book and plan to use it again for future graduate-level teaching in statistics."
—Journal of the American Statistical Association
"This book has tremendous potential for usage in statistics and biostatistics departments where the Ph.D. students would not necessarily have taken a measure theory course but would need a rigorous treatment of probability for their dissertation research and publications in statistical and biostatistics journals … The authors are commended for providing this valuable book for students in statistics and biostatistics. The illustrative biostatistics examples (throughout chapter 10 but especially in chapter 11) provide motivating rewards for students."
—Robert Taylor, Clemson University
"… a very good textbook choice for our courses on advanced probability theory (I, II) at the graduate level."
—Jie Yang, University of Illinois at Chicago
"Many successful graduate students in statistics lack the mathematical prerequisites necessary for Billingsley’s book and find su