StatlectThe Digital Textbook

Fundamentals of probability

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.

Probability and events

Zero-probability events

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


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

Bayes' rule

Prior probability, posterior probability, updating

Conditional probability

How to revise probabilities when new information arrives

Independent events

Definition and explanation of independence and mutual independence

Random variables and random vectors

Random vectors

Joint distributions, marginal distributions

Random variables

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

Expected value

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

Linear correlation

Another measure of association between random variables


Association between random variables, definition, interpretation, properties

Indicator functions

Equal to one when an event happens and zero otherwise

Covariance matrix

A multivariate generalization of the concept of variance

Conditional probabilities, conditional distributions and independence

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

Independent random variables

Definition and characterization of independence between random variables

Conditional expectation

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

Multiple choice tests

Test 2

Random variables and random vectors

Test 1

Probability, conditional probability, independent events

Test 3

Expected value, variance, covariance

The book

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