Introduction to Probability

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

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