2nd Edition

# Introduction to Probability, Second Edition

By Joseph K. Blitzstein, Jessica Hwang Copyright 2019
634 Pages
by Chapman & Hall

634 Pages
by Chapman & Hall

Also available as eBook on:

Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory.

The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces.

The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment.

The second edition adds many new examples, exercises, and explanations, to deepen understanding of the ideas, clarify subtle concepts, and respond to feedback from many students and readers. New supplementary online resources have been developed, including animations and interactive visualizations, and the book has been updated to dovetail with these resources.

Supplementary material is available on Joseph Blitzstein’s website www. stat110.net. The supplements include:
Solutions to selected exercises
Handouts including review material and sample exams Animations and interactive visualizations created in connection with the edX online version of Stat 110.
Links to lecture videos available on ITunes U and YouTube There is also a complete instructor's solutions manual available to instructors who require the book for a course.

1. Probability and Counting
2. 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

3. Conditional Probability
4. 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

Recap

R

Exercises

5. Random Variables and Their Distributions
6. Random variables

Distributions and probability mass functions

Bernoulli and Binomial

Hypergeometric

Discrete Uniform

Cumulative distribution functions

Functions of random variables

Independence of rvs

Connections between Binomial and Hypergeometric

Recap

R

Exercises

7. Expectation
8. Definition of expectation

Linearity of expectation

Geometric and Negative Binomial

Indicator rvs 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

9. Continuous Random Variables
10. Probability density functions

Uniform

Universality of the Uniform

Normal

Exponential

Poisson processes

Symmetry of iid continuous rvs

Recap

R

Exercises

11. Moments
12. Summaries of a distribution

Interpreting moments

Sample moments

Moment generating functions

Generating moments with MGFs

Sums of independent rvs via MGFs

*Probability generating functions

Recap

R

Exercises

13. Joint Distributions
14. Joint, marginal, and conditional

D LOTUS

Covariance and correlation

Multinomial

Multivariate Normal

Recap

R

Exercises

15. Transformations
16. Change of variables

Convolutions

Beta

Gamma

Beta-Gamma connections

Order statistics

Recap

R

Exercises

17. Conditional Expectation
18. Conditional expectation given an event

Conditional expectation given an rv

Properties of conditional expectation

*Geometric interpretation of conditional expectation

Conditional variance

Recap

R

Exercises

19. Inequalities and Limit Theorems
20. Inequalities

Law of large numbers

Central limit theorem

Chi-Square and Student-t

Recap

R

Exercises

21. Markov Chains
22. Markov property and transition matrix

Classification of states

Stationary distribution

Reversibility

Recap

R

Exercises

23. Markov Chain Monte Carlo
24. Metropolis-Hastings

Recap

R

Exercises

25. Poisson Processes

Poisson processes in one dimension

Conditioning, superposition, thinning

Poisson processes in multiple dimensions

Recap

R

Exercises

A Math

A Sets

A Functions

A Matrices

A Difference equations

A Differential equations

A Partial derivatives

A Multiple integrals

A Sums

A Pattern recognition

A Common sense and checking answers

B R

B Vectors

B Matrices

B Math

B Sampling and simulation

B Plotting

B Programming

B Summary statistics

B Distributions

C Table of distributions

Bibliography

Index

### Biography

Joseph K. Blitzstein, PhD, professor of the practice in statistics, Department of Statistics, Harvard University, Cambridge, Massachusetts, USA

Jessica Hwang is a graduate student in the Stanford statistics department.