Expected value is one of the fundamental concepts in probability, in a sense more general than probability itself. The expected value of a real-valued random variable gives a measure of the center of the distribution of the variable. More importantly, by taking the expected value of various functions of a general random variable, we can measure many interesting features of its distribution, including spread, skewness, kurtosis, and correlation. Generating functions are certain types of expected value that completely determine the distribution of the variable. Conditional expected value, which incorporates known information in the computation, is one of the fundamental concepts in probability.

In the advanced topics, we define expected value as an integral with respect to the underlying probability measure. We also revisit conditional expected value from a measure-theoretic point of view. We study vector spaces of random variables with certain expected values as the norms of the spaces, which in turn leads to modes of convergence for random variables.

- Definitions and Basic Properties
- Additional Properties
- Variance
- Skewness and Kurtosis
- Covariance and Correlation
- Generating Functions
- Conditional Expected Value

- Expected Value and Covariance Matrices
- Expected Value as an Integral
- Conditional Expected Value Revisited
- Vector Spaces of Random Variables
- Uniformly Integrable Variables
- Kernels and Operators

- Dice Experiment
- Special Distribution Simulator
- Interactive Histogram
- Error Function App
- Bivariate Uniform Experiment
- Die-Coin Experiment
- Coin-Die Experiment

- 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 via Expectation. Peter Whittle (1970)
- Probability: Theory and Examples. Richard Durrett
- Wikipedia articles on probability
- Wolfram MathWorld articles on probability and statistics

Suam habet fortuna rationem

(Chance has its reason)—Petronius