Author: Capri Granville

A free online version of the second edition of the book based on Stat 110, *Introduction to Probability* by Joe Blitzstein and Jessica Hwang, is now available here. Print copies are available via CRC Press, Amazon, and elsewhere. Stat110x is also available as an free edX course, here.

The edX course focuses on animations, interactive features, readings, and problem-solving, and is **complementary** to the Stat 110 lecture videos on YouTube, which are available here. The Stat110x animations are available within the course and here. For more information, visit Stat110 at Harvard. A 10-page cheat sheet summarizing the content, is available here. For more free books, visit this page.

**Table of Contents**

**1. 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

**2. 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

**3. 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 rvs
- Connections between Binomial and Hypergeometric
- Recap
- R
- Exercises

**4. Expectation**

- 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

**5. Continuous Random Variables**

- Probability density functions
- Uniform
- Universality of the Uniform
- Normal
- Exponential
- Poisson processes
- Symmetry of iid continuous rvs
- Recap
- R
- Exercises

**6. Moments**

- 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

**7. Joint Distributions**

- Joint, marginal, and conditional
- D LOTUS
- Covariance and correlation
- Multinomial
- Multivariate Normal
- Recap
- R
- Exercises

**8. Transformations**

- Change of variables
- Convolutions
- Beta
- Gamma
- Beta-Gamma connections
- Order statistics
- Recap
- R
- Exercises

**9. Conditional Expectation**

- 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

10. Inequalities and Limit Theorems

Inequalities

Law of large numbers

Central limit theorem

Chi-Square and Student-t

Recap

R

Exercises

**11. Markov Chains**

- Markov property and transition matrix
- Classification of states
- Stationary distribution
- Reversibility
- Recap
- R
- Exercises

**12. Markov Chain Monte Carlo**

- Metropolis-Hastings
- Recap
- R
- Exercises

**13. Poisson Processes**

- Poisson processes in one dimension
- Conditioning, superposition, thinning
- Poisson processes in multiple dimensions
- Recap
- R
- Exercises

**A Math**

- Sets
- Functions
- Matrices
- Difference equations
- Differential equations
- Partial derivatives
- Multiple integrals
- Sums
- Pattern recognition
- Common sense and checking answers

**B R Programming**

- Vectors
- Matrices
- Math
- Sampling and simulation
- Plotting
- Programming
- Summary statistics
- Distributions

**C Table of distributions**

**Bibliography**

**Index**