Recall that a probability distribution is just another name for a probability measure. Most distributions are associated with random variables, and in fact every distribution *can* be associated with a random variable. In this chapter we explore the basic types of probability distributions (discrete, continuous, mixed), and the ways that distributions can be defined using density functions, distribution functions, and quantile functions. We also study the relationship between the distribution of a random vector and the distributions of its components, conditional distributions, and how the distribution of a random variable changes when the variable is transformed.

In the advanced sections, we study convergence in distribution, one of the most important types of convergence. We also construct the abstract integral with respect to a positive measure and study the basic properties of the integral. This leads in turn to general (signed measures), absolute continuity and singularity, and the existence of density functions. Finally, we study various vector spaces of functions that are defined by integral properties.

- Discrete Distributions
- Continuous Distributions
- Mixed Distributions
- Joint Distributions
- Conditional Distributions
- Distribution and Quantile Functions
- Transformations of Random Variables

- Convergence in Distribution
- General Distribution Functions
- The Integral With Respect to a Measure
- Properties of the Integral
- General Measures
- Absolute Continuity and Density Functions
- Function Spaces

- Binomial Coin Experiment
- Dice Experiment
- Die-Coin Experiment
- Coin-Die Experiment
- Special Distribution Simulator
- Special Distribution Calculator
- Random Quantile Experiment
- Rejection Method Experiment
- Bivariate Uniform Experiment
- Histogram App

- An Introduction to Probability Theory and Its Applications. William Feller
- A First Course in Probability. Sheldon Ross
- The Essentials of Probability. Richard Durrett
- Probability and Measure. Patrick Billingsley
- Probability: Theory and Examples. Richard Durrett
- Wikipedia articles on probability
- Wolfram MathWorld articles on probability and statistics
- Wikipedia articles on probability distributions

Nothing can permanently please which does not contain in itself the reason why it is so and not otherwise.

—Samuel Taylor Coleridge