### Summary

**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 **

Why More Rigor Is Needed

**
****Size Matters**

Cardinality

Summary

**
****The Elements of Probability Theory**

Introduction

Sigma-Fields

The Event That *A*_{n} Occurs Infinitely Often

Measures/Probability Measures

Why Restriction of Sets Is Needed

When We Cannot Sample Uniformly

The Meaninglessness of Post-Facto Probability Calculations

Summary

**
****Random Variables and Vectors**

Random Variables

Random Vectors

The Distribution Function of a Random Variable

The Distribution Function of a Random Vector

Introduction to Independence

Take (Ω, *F, P*) = ((0, 1), *B*_{(0,1)}, *μ*_{L}), Please!

Summary

**
****Integration and Expectation**

Heuristics of Two Different Types of Integrals

Lebesgue–Stieltjes Integration

Properties of Integration

Important Inequalities

Iterated Integrals and More on Independence

Densities

Keep It Simple

Summary

**
****Modes of Convergence**

Convergence of Random Variables

Connections between Modes of Convergence

Convergence of Random Vectors

Summary

**
****Laws of Large Numbers**

Basic Laws and Applications

Proofs and Extensions

Random Walks

Summary

**
****Central Limit Theorems**

CLT for iid Random Variables and Applications

CLT for Non iid Random Variables

Harmonic Regression

Characteristic Functions

Proof of Standard CLT

Multivariate Ch.f.s and CLT

Summary

**
****More on Convergence in Distribution**

Uniform Convergence of Distribution Functions

The Delta Method

Convergence of Moments: Uniform Integrability

Normalizing Sequences

Review of Equivalent Conditions for Weak Convergence

Summary

**
****Conditional Probability and Expectation **

When There Is a Density or Mass Function

More General Definition of Conditional Expectation

Regular Conditional Distribution Functions

Conditional Expectation as a Projection

Conditioning and Independence

Sufficiency

Expect the Unexpected from Conditional Expectation

Conditional Distribution Functions as Derivatives

Appendix: Radon–Nikodym Theorem

Summary

**
****Applications**

*F(X)* ~ *U*[0, 1] and Asymptotics

Asymptotic Power and Local Alternatives

Insufficient Rate of Convergence in Distribution

Failure to Condition on All Information

Failure to Account for the Design

Validity of Permutation Tests: I

Validity of Permutation Tests: II

Validity of Permutation Tests III

A Brief Introduction to Path Diagrams

Estimating the Effect Size

Asymptotics of an Outlier Test

An Estimator Associated with the Logrank Statistic

**
**Appendix A: Whirlwind Tour of Prerequisites

Appendix B: Common Probability Distributions

Appendix C: References

Appendix D: Mathematical Symbols and Abbreviations

Index

### Author(s) Bio

**Michael A. Proschan** is a mathematical statistician in the Biostatistics Research Branch at the U.S. National Institute of Allergy and Infectious Diseases (NIAID). A fellow of the American Statistical Association, Dr. Proschan has published more than 100 articles in numerous peer-reviewed journals. His research interests include monitoring clinical trials, adaptive methods, permutation tests, and probability. He earned a PhD in statistics from Florida State University.

**Pamela A. Shaw** is an assistant professor of biostatistics in the Department of Biostatistics and Epidemiology at the University of Pennsylvania Perelman School of Medicine. Dr. Shaw has published several articles in numerous peer-reviewed journals. Her research interests include methodology to address covariate and outcome measurement error, the evaluation of diagnostic tests, and the design of medical studies. She earned a PhD in biostatistics from the University of Washington.

### Reviews

" 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 such a course too hard … The strong points of this book are a good selection of topics, good choices for proofs to include and omit, and interesting examples. Some of the examples motivate the need for mathematical theory while others illustrate the relation of the theory to statistical practice. When there is a need for it, the presentation of the material includes side explanations that should help a student with a less solid math background."

—Wlodek Byrc, University of Cincinnati

"I think the authors have done a great job at writing this book. The material is presented carefully and the examples and exercises are appropriate and extremely helpful … The strongest point of this book is the large collection of statistical applications presented along with each topic. These examples are used to motivate the need for fundamental probabilistic results. The authors have also included a great set of exercises at the end of each section. Another good idea is the summary presented at the end of each chapter. … I would be happy to adopt this book as a required text for the course that I teach; its content is appropriate and at the right level."

—Radu Herbei, Ohio State University