This is an introduction to the main concepts of probability theory. After reading the lectures, test your knowledge with the multiple choice tests in the last section.

Events having zero probability, almost sure events, almost sure properties

Sample space, sample points, events, probability and its properties

Prior probability, posterior probability, updating

How to revise probabilities when new information arrives

Definition and explanation of independence and mutual independence

Joint distributions, marginal distributions

Discrete and continuous random variables, probability mass and density functions

Expected value and the Lebesgue integral

A rigorous definition of expected value, based on the Lebesgue integral

The mean of a random variable, how to compute it, its properties

Dispersion around the mean, definition, interpretation, properties

Properties of the expected value

Linearity of the expected value, expectation of positive random variables, other properties

Another measure of association between random variables

Association between random variables, definition, interpretation, properties

Equal to one when an event happens and zero otherwise

A multivariate generalization of the concept of variance

Conditional probability distributions

How to update the distribution of a random variable after receiving some information

Conditional probability as a random variable

A more rigorous presentation of conditional probability

Definition and characterization of independence between random variables

How to compute the expected value of a random variable after observing the value of another

Random variables and random vectors

Probability, conditional probability, independent events

Expected value, variance, covariance

The book

Most learning materials found on this website are now available in a traditional textbook format.

Featured pages

- Central Limit Theorem
- Binomial distribution
- Combinations
- Normal distribution
- Mean square convergence
- Multinomial distribution

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Main sections

- Mathematical tools
- Fundamentals of probability
- Additional topics in probability
- Probability distributions
- Asymptotic theory
- Fundamentals of statistics

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Glossary entries

- Mean squared error
- Continuous random variable
- Type II error
- Jensen inequality
- Factorial
- Chebyshev inequality

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