Probability

The aim of this course is to provide an introduction to the basic concepts of probability calculus and the modeling of random phenomena.
Course outline:



Terminology and notation


Probability space
2.1 Tribe and events
2.2 Probability
2.3 Independence of events
2.4 Conditional probability
2.5 Limit of series of sets


Random variables
3.1 Random variable (discrete and density)
3.2 Law of a random variable.
3.3 Expectation of a random variable
3.4 Independence of random variables
3.5 Properties of expectation
3.6 Variance and Covariance
3.7 Tools for random variable laws
3.8 Gaussian vectors
3.9 Conditional expectation


Convergence of random variable sequences
4.1 Different modes of convergence
4.2 Laws of large numbers
4.3 Convergence into a law
4.4 The central limit theorem
4.5 Monte Carlo method

Basic notions of analysis.

Terminology and notation


Probability space
2.1 Tribe and events
2.2 Probability
2.3 Independence of events
2.4 Conditional probability
2.5 Limit of sequence of sets


Random variables
3.1 Random variable (discrete and density)
3.2 Law of a random variable.
3.3 Expectation of a random variable
3.4 Independence of random variables
3.5 Properties of expectation
3.6 Variance and Covariance
3.7 Tools for random variable laws
3.8 Gaussian vectors
3.9 Conditional expectation


Convergence of random variable sequences
4.1 Different modes of convergence
4.2 Laws of large numbers
4.3 Convergence into a law
4.4 The central limit theorem
4.5 Monte Carlo method