Introduction to Probability, Second Edition

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 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 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.