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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.
Introduction
- Why More Rigor Is Needed
Size Matters
- Cardinality
- Summary
The Elements of Probability Theory
- Introduction
- Sigma-Fields
- The Event That An 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), µL), 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
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.
"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
