ENS Paris-Saclay

The introductory statistics course presents the fundamental concepts of descriptive and inferential statistics, enabling the collection, organization, analysis, and interpretation of numerical data.

It introduces students to the main methods for summarizing data, evaluating relationships between variables, and performing estimations and hypothesis tests.

The course covers the formal definition and construction of a Markov chain, fundamental properties such as the Markov property, absorbing states, and communication classes.

It also includes key theorems like the ergodic theorem, convergence to a stationary distribution, and absorption times. This course prepares students for in-depth analysis of chains and their various applications in probability, statistics, and more generally in data science.

The objective of this course is to deepen statistical inference methods, including point and interval estimation, hypothesis testing, and analysis of variance (ANOVA). It also covers simple linear models, multivariate statistics, and basic stochastic processes.

This course combines theory and practical applications using statistical software, enabling students to design, analyze, and interpret quantitative studies in biology or economics.

This course introduces the foundations of measure and integration theory, with particular attention given to the construction of σ-algebras, the notion of measurability, and the Lebesgue integral. It explores functional spaces and their properties, in connection with the tools of modern analysis.

Applications to probability illustrate the scope of these concepts in modeling random phenomena and in the rigorous treatment of random variables.

This course offers an introduction to signal theory, in particular to the discrete Fourier transform and the Nyquist-Shannon theorem, which make it possible to analyze and reconstruct digital signals.

It then covers the fundamentals of discrete optimization and convex optimization, with or without constraints, emphasizing analytical and algorithmic solution methods.

This course deals with the solution of ordinary differential equations (ODEs) of order 1 and 2, as well as linear differential systems of order 1 and their extension to ODEs of order n. It covers the Cauchy-Lipschitz theorem, qualitative analysis of solutions, and offers an introduction to partial differential equations (PDEs).

The program provides the essential knowledge needed to understand and use the main tools of differential analysis.

This course in Data Analysis and Modeling for Sciences introduces fundamental data analysis and modeling methods common to all scientific disciplines: basic sciences, natural sciences, humanities and social sciences. It covers data collection, visualization, and statistical analysis, as well as mathematical modeling and protocol modeling.

The course utilizes languages such as R, Python, and Julia to develop students' analytical and programming skills.