Université Paris 1 Panthéon-Sorbonne

Number of Credits: 7 credits

Hours: 30 hours of Lectures, 30 hours of Tutorials including exam, and 20 hours of flipped Classrooms with tutor support.

General Presentation: This course introduces the student to the fundamentals of rigorous probability theory.

Part 1: is intended to be an introductory course in elementary probability theory. Before introducing probability it will begin by looking at elementary set theory and then combinatorial analysis. Then it will introduce the concept of a probability measure, sigma-algebras, and the axioms of probability. Students will study conditional probability and independence. Then students will define and analyze random variables in the discrete case, furthering this route by studying several usual discrete random variables. Finally the course will present random variables in the continuous case, the usual continuous random variables.

Part 2: In this second part, we  first study the joint distribution of pair of continuous random variables as an application of the change of variable theorem. Then, we introduce the notion of conditional distribution in order to be able to define in our setting the conditional expectation. After that we will study the convergence of sequences of random variables and more precisely the weak law of large numbers  and the central limit theorem. Finally as an application,  we will introduce the primer of estimation theory studying in particular the method of moments and the maximum likelihood estimators.

Books: John Rice's book entitled, Mathematical Statistics and Data Analysis, 3rd edition. The 8 first chapters.

Prerequisites: Logic and Sets