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
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
Random variables and random vectors
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 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
Definition and characterization of independence between random variables
How to compute the expected value of a random variable after observing the value of another
Multiple choice tests
Random variables and random vectors
Probability, conditional probability, independent events
Expected value, variance, covariance
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