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Fundamentals of probability theory

This is an introduction to the fundamental concepts of probability theory. Each lecture contains detailed proofs and derivations of all the main results, as well as solved exercises.

Probability and events

Zero-probability events

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


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

Bayes' rule is one of the fundamental formulae of probability theory.

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 theory of Lebesgue integration

Expected value

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


Dispersion around the mean, definition, interpretation, fundamental 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


Cut-off point of a distribution that leaves to its left a given proportion of the distribution

A fundamental formula used to compute the variance of a random variable.

Conditional distributions and independence

Conditional probability distributions

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

Rigorous conditional probability

A more rigorous presentation of conditional probability based on the theory of sigma-algebras

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


Chebyshev's inequality

A fundamental inequality derived from Markov's inequality

Markov's inequality

Provides an upper bound to the probability that a random variable will exceed a threshold

Jensen's inequality

Concerns the expected value of convex and concave transformations of a random variable

Three fundamental inequalitites: Markov's, Chebyshev's and Jensen's.

More about probability mass and density functions

Legitimate probability density functions

Properties of probability density functions and how to construct them

Legitimate probability mass functions

Properties of probability mass functions and how to construct them

Factorization of joint pdfs

Factorization into marginal and conditional probability density function

Factorization of joint pmfs

Factorization into marginal and conditional probability mass function

Transformations of random variables

Functions of random vectors

How to derive the joint distribution of a function of a random vector

Functions of random variables

How to derive the distribution of Y=g(X) from the distribution of X

Sums of independent random variables

How to derive the distribution of a sum from the distributions of the summands

Convolution formulae are used to compute the distribution of a sum of two random variables.



Definition of cross-moment and central cross-moment of a random vector


Definition of moment and central moment of a random variable

Moment generating and characteristic functions

Joint moment generating function

Generalizes the concept of moment generating function to random vectors

Moment generating function

Definition, computation of moments, characterization of distributions

Joint characteristic function

Generalizes the concept of characteristic function to random vectors

Characteristic function

Definition, computation of moments, characterization of distributions

Cumulant generating function

The logarithm of the moment generating function has some interesting properties

The moment generating function is a fundamental function that allows us to completely characterize a probability distribution and to derive its moments.

Information-theoretic concepts

Kullback-Leibler divergence

A measure of the dissimilarity between two probability distributions

Revise what you studied

Probability questions

200 probability questions to check your knowledge of probability theory

The books

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