Introduction to Probability

Joseph K. Blitzstein and Jessica Hwang

Publisher:

Chapman & Hall/CRC

Publication Date:

2014

Number of Pages:

580

Format:

Hardcover

Series:

Texts in Statistical Science

Price:

99.95

ISBN:

9781466575578

Category:

Textbook

MAA Review
Table of Contents

[Reviewed by

Peter Rabinovitch

, on

08/24/2015

]

Introduction to Probability is a very nice text for a calculus-based first course in probability. There are many features that make this book attractive as a text for such a course.

Each chapter starts off with an introductory section, followed by detail sections expanding on the introduction. Chapters conclude with a summary, a small section with R code to explore some of the topics within the chapter, and an extensive problem section which is divided into parts for each of the sections of the chapter, and a “mixed practice” group of problems that may require ideas from the whole chapter or from previous parts of the book.

The exercises are truly impressive. There are about 600 and some of them are very interesting and new to me. For example, I had not previously seen problem 83 in chapter 4 on Feynman’s approach to deciding what to eat in a restaurant. Some of the problems are rote, but many include interesting asides and commentary. In addition, about 250 of the problems have solutions on the stat110.net web site, which is another feature that will appeal to students. The web site has R code, the previously mentioned solutions, and many videos from the authors teaching the class. The videos are entertaining as well as informative.

The authors stress the importance of understanding throughout, with “story proofs,” clear explanations, and warnings to the students of tricky points.

In addition to the standard material for such a course, there are also very nicely done chapters on inequalities and limit theorems, Markov chains, and Markov chain Monte Carlo. The only chapter I found disappointing at all was the last on an introduction to Poisson processes, which seemed a little too standard in comparison to the previous chapters.

That small disappointment aside, this is an excellent text and deserves serious consideration. Checking out stat110.net will surely help you decide if this book is right for you.

Peter Rabinovitch has been doing data science since long before “data science” was a thing.

Probability and Counting
Why Study Probability?
Sample Spaces and Pebble World
Naive Definition of Probability
How to Count
Story Proofs
Non-Naive Definition of Probability
Recap
R
Exercises

Conditional Probability
The Importance of Thinking Conditionally
Definition and Intuition
Bayes’ Rule and the Law of Total Probability
Conditional Probabilities Are Probabilities
Independence of Events
Coherency of Bayes’ Rule
Conditioning as a Problem-Solving Tool
Pitfalls and Paradoxes
Recap
R
Exercises

Random Variables and Their Distributions
Random Variables
Distributions and Probability Mass Functions
Bernoulli and Binomial
Hypergeometric
Discrete Uniform
Cumulative Distribution Functions
Functions of Random Variables
Independence of r.v.s
Connections Between Binomial and Hypergeometric
Recap
R
Exercises

Expectation
Definition of Expectation
Linearity of Expectation
Geometric and Negative Binomial
Indicator r.v.s and the Fundamental Bridge
Law of The Unconscious Statistician (LOTUS)
Variance
Poisson
Connections Between Poisson and Binomial
Using Probability and Expectation to Prove Existence
Recap
R
Exercises

Continuous Random Variables
Probability Density Functions
Uniform
Universality of The Uniform
Normal
Exponential
Poisson Processes
Symmetry of i.i.d. Continuous r.v.s
Recap
R
Exercises

Moments
Summaries of a Distribution
Interpreting Moments
Sample Moments
Moment Generating Functions
Generating Moments With MGFs
Sums of Independent r.v.s Via MGFs
Probability Generating Functions
Recap
R
Exercises

Joint Distributions
Joint, Marginal, and Conditional
2D LOTUS
Covariance and Correlation
Multinomial
Multivariate Normal
Recap
R
Exercises

Transformations
Change of Variables
Convolutions
Beta
Gamma
Beta-Gamma Connections
Order Statistics
Recap
R
Exercises

Conditional Expectation
Conditional Expectation Given an Event
Conditional Expectation Given an r.v.
Properties of Conditional Expectation
Geometric Interpretation of Conditional Expectation
Conditional Variance
Adam and Eve Examples
Recap
R
Exercises

Inequalities and Limit Theorems
Inequalities
Law of Large Numbers
Central Limit Theorem
Chi-Square and Student-t
Recap
R
Exercises

Markov Chains
Markov Property and Transition Matrix
Classification of States
Stationary Distribution
Reversibility
Recap
R
Exercises

Markov Chain Monte Carlo
Metropolis-Hastings
Gibbs Sampling
Recap
R
Exercises

Poisson Processes
Poisson Processes in One Dimension
Conditioning, Superposition, Thinning
Poisson Processes in Multiple Dimensions
Recap
R
Exercises

Math
Sets
Functions
Matrices
Difference Equations
Differential Equations
Partial Derivatives
Multiple Integrals
Sums
Pattern Recognition
Common Sense and Checking Answers

R
Vectors
Matrices
Math
Sampling and Simulation
Plotting
Programming
Summary Statistics
Distributions

Table of Distributions

Bibliography

Index