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# Introduction to Probability, Second Edition

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## Book Description

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

Additional practice problems

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.

## Table of Contents

- Probability and Counting
- Conditional Probability
- Random Variables and Their Distributions
- Expectation
- Continuous Random Variables
- Moments
- Joint Distributions
- Transformations
- Conditional Expectation
- Inequalities and Limit Theorems
- Markov Chains
- Markov Chain Monte Carlo
- Poisson Processes

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

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

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

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

Probability density functions

Uniform

Universality of the Uniform

Normal

Exponential

Poisson processes

Symmetry of iid continuous rvs

Recap

R

Exercises

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

Joint, marginal, and conditional

D LOTUS

Covariance and correlation

Multinomial

Multivariate Normal

Recap

R

Exercises

Change of variables

Convolutions

Beta

Gamma

Beta-Gamma connections

Order statistics

Recap

R

Exercises

Conditional expectation given an event

Conditional expectation given an rv

Properties of conditional expectation

*Geometric interpretation of conditional expectation

Conditional variance

Adam and Eve examples

Recap

R

Exercises

Inequalities

Law of large numbers

Central limit theorem

Chi-Square and Student-t

Recap

R

Exercises

Markov property and transition matrix

Classification of states

Stationary distribution

Reversibility

Recap

R

Exercises

Metropolis-Hastings

Recap

R

Exercises

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

## Author(s)

### 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.