Discrete mathematics and probability

Teacher(s)

. SOMEBODY; Delvenne Jean-Charles; Pereira Olivier;

Language

French

Prerequisites

This course assumes acquired the basic notion of mathematics (analysis) such as taught in the course LEPL1102

Main themes

The course presents the fundamental concepts of discrete mathematics (counting, and graph theory) as well as probabilities necessary for engineering disciplines (random variables, conditional probability, dependence between random variables, estimation and limit theorems).

Learning outcomes

At the end of this learning unit, the student is able to :

At the end of this course, the student will be able to: understand and use basic counting concepts, including using recurrent formulas; make the link between the concepts of counting and those of injection, surjection and bijection; master the elements of modular arithmetic within the framework of an application, according to the time available; use the basic concepts of graph theory; define, describe, explain and use the concepts of discrete and continuous, univariate and bivariate random variables; measure the dependence between two random variables; estimate the characteristics of random variables (expectation, variance, covariance, ...), and make the difference between these parameters of the population and their estimation; use Chebycheff's inequality and limit theorems to characterize random variables; reformulate the textual statement of a problem in an unambiguous mathematical and probabilistic formalism, using the appropriate theoretical concepts and tools; solve an applied problem by following a deductive approach based on the correct and useful manipulation of expressions; validate the internal consistency of the formalization and the solution of a probability calculation problem on the basis of the logical constraints induced by the theory.

Content

Discrete Mathematics:
o Combinatorics and counting
o Link between counting and injections, surjections and bijections
o Elements of graph theory
o Elements of modular arithmetic (including introduction to cryptography or
error correcting codes)
Probabilities
o Introduction to statistical data modeling and probability concepts in engineering contexts
o Events and probabilities, particularly in relation to combinatorics
o Random variables: discrete and continuous (univariate), including pdf and cdf
o Examples of random variables: Binomial, Poisson, Gaussian, exponential
o Bivariate random variables (discrete and continuous)
o Joint distributions, marginal and conditional distributions, independence
o Study of the characteristics of uni- and bivariate distributions via simulations on
computer
o Mean, variance, covariance and correlation, expectation and conditional variance
o Introduction to the estimation of these characteristic quantities
o Law of Large Numbers and Central Limit Theorem

Teaching methods

The course will consist of:
- ex cathedra presentations which will present the concepts and tools on the basis of examples from the engineering world;
- exercise sessions (APE) aimed at systematically putting into practice the different notions structured during the course.
- case studies (APP) which will give the student the opportunity to discover certain notions through problems.
Homework and mini-projects may also be offered and their evaluation will not contribute to the final grade.
Examples related to sustainable development and transition will be evoked.

Evaluation methods

Written exam during the session. An oral examination may also be required, under specific individual circumstances.

Online resources

The Moodle page of the course.

Teaching materials

• Documents sur la page Moodle / Documents on the Moodle page

Faculty or entity

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