Mathematics - Probabilities

ECTS

SEMESTER

lectures

classes / seminars

practical work

integrative teaching

independent work

10h

20h

0h

0h

30h

Language used

French

Course supervisor(s)

Key words

probability, measure theory, Lebesgue integral, random variable, law of large numbers, central limit theorem, Gaussian vectors

Prerequisites

basic knowledge of calculus, linear algebra, Riemann integral and elementary set theory

The objective of the course is to master classical notions in probability theory, in measure theory, and to know how to compute the law of some random variables

This course is concerned with classical notions in probability and measure theory, which include the following topics:

1. Probability spaces, random variable and expectation.

2. Discrete and absolutely continuous distributions, classical distributions, Law of the unconscious statistician.

3. Characterization of laws through the cumulative distribution and characteristic functions. Computation of the law of a random variable.

4. Random vectors, marginal laws, Fubini's theorem, Change of variable

5. Fatou's Lemma, Monotone convergence theorem, Dominated convergence theorem

6. Lp spaces, moments, variance, covariance, linear regression

7. Independance and Convolution

8. Different notions of convergence of random variables : in probability, in distribution, almost sure, in Lp. Borel Cantelli Lemma.

9. Gaussian vectors

Students will also be provided with the basic tools to numerically simulate random variables in Python and Matlab.

Levels

Description and operational verbs

Know

The fundamental aspects of measure theory and the basic tools for computation of probability distributions

Understand

To understand the logic of probability theory and the way it applies to concrete calculous

Apply 

To deploy calculus methods to resolve problems in probability, choosing among different tools to describe random behaviours

Analyse 

To detect basic properties of random phenomena and to devise a mathematical strategy to analyse them

Summarise

To elaborate a well structured discussion about topics on random phenomena

Assess

To detect aberration in probability results and assess the validity of different property concerning random phenomena, as independance, convergence or integrability

Continuous assessment

Written test

Oral presentation / viva

Written report / project