Probability

Objectives

This course will enable students to master the basic concepts of probability calculus, which will be central to all statistics courses at ENSAI. In particular, students will be able to

Use appropriate techniques to calculate probability laws and mathematical expectations, in one or more dimensions,

Identify common probability laws and their characteristic properties,

Determine and use the properties of independence between several random variables.

Course outline

Random variables: definition, notion of probability law, discrete or density laws, mathematical expectation, transfer theorem. Moments and Lp spaces, distribution function, Laplace transform, characteristic function. Methods for calculating probability distributions and probabilistic inequalities.

Random vectors and multivariate laws: notion of probabilistic independence, characterization of independence, convolution. Notion of covariance and correlation. Gaussian vectors.

Prerequisites

Integral calculus and measure theory, notions of linear algebra, discrete probabilities