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.
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.
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 ( , F, P) = ((0, 1), B(0,1), 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