Essentials of Probability Theory for Statisticians

Michael A. Proschan, Pamela A. Shaw



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 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