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 TCSS6AA

Mathematics 2: Probability


Duration : 30 hours

ECTS Credits : 3.5

Semester: S5

Person(s) in charge:

 Denis Villemonais, Associate professor, denis.villemonais@mines-nancy.univ-lorraine.fr 

 

Keywords : Probability theory, probability distribution, random variable

 

Prerequisite :

Familiarity with basic notions of analysis, topology and algebra. Familiarity with rigorous proofs.

 

Aims :  Students completing this course will be able to understand and manipulate classical notions in probability and measure theory.

 

Contents

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.

 

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

Skills: 

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 calculous tecnics 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

Evaluations :

  • Written test
  • Continuous Control
  • Oral report
  • Project
  • Written Report
  • Aucune étiquette