The Probability Lifesaver: All the Tools You Need to Understand Chance

The essential lifesaver for students who want to master probability

For students learning probability, its numerous applications, techniques, and methods can seem intimidating and overwhelming. That’s where The Probability Lifesaver steps in. Designed to serve as a complete stand-alone introduction to the subject or as a supplement for a course, this accessible and user-friendly study guide helps students comfortably navigate probability’s terrain and achieve positive results.

The Probability Lifesaver is based on a successful course that Steven Miller has taught at Brown University, Mount Holyoke College, and Williams College. With a relaxed and informal style, Miller presents the math with thorough reviews of prerequisite materials, worked-out problems of varying difficulty, and proofs. He explores a topic first to build intuition, and only after that does he dive into technical details. Coverage of topics is comprehensive, and materials are repeated for reinforcement—both in the guide and on the book’s website. An appendix goes over proof techniques, and video lectures of the course are available online. Students using this book should have some familiarity with algebra and precalculus.

The Probability Lifesaver not only enables students to survive probability but also to achieve mastery of the subject for use in future courses.

• A helpful introduction to probability or a perfect supplement for a course
• Numerous worked-out examples
• Lectures based on the chapters are available free online
• Intuition of problems emphasized first, then technical proofs given
• Appendixes review proof techniques
• Relaxed, conversational approach

Table of Contents

Part I General Theory
Chapter 1 Introduction
Chapter 2 Basic Probability Laws
Chapter 3 Counting I: Cards
Chapter 4 Conditional Probability, Independence, And Bayes’ Theorem
Chapter 5 Counting Ii: Inclusion-Exclusion
Chapter 6 Counting Iii: Advanced Combinatorics

Part II Introduction To Random Variables
Chapter 7 Introduction To Discrete Random Variables
Chapter 8 Introduction To Continuous Random Variables
Chapter 9 Tools: Expectation
Chapter 10 Tools: Convolutions And Changing Variables
Chapter 11 Tools: Differentiating Identities

Part III Special Distributions
Chapter 12 Discrete Distributions
Chapter 13 Continuous Random Variables: Uniform And Exponential
Chapter 14 Continuous Random Variables: The Normal Distribution
Chapter 15 The Gamma Function And Related Distributions
Chapter 16 The Chi-Square Distribution

Part IV Limit Theorems
Chapter 17 Inequalities And Laws Of Large Numbers
Chapter 18 Stirling’S Formula
Chapter 19 Generating Functions And Convolutions
Chapter 20 Proof Of The Central Limit Theorem
Chapter 21 Fourier Analysis And The Central Limit Theorem

Part V Additional Topics
Chapter 22 Hypothesis Testing
Chapter 23 Difference Equations, Markov Processes, And Probability
Chapter 24 The Method Of Least Squares
Chapter 25 Two Famous Problems And Some Coding

Appendix A Proof Techniques
Appendix B Analysis Results
Appendix C Countable and Uncountable Sets
Appendix D Complex Analysis and the Central Limit Theorem