The goal of this course is to introduce the student to the formal way to deal with uncertainty. The course aims at being rigorous, and complete although the technicalities will be contained. The course will cover the basic chapters of probability theory, with an introduction to martingales and their properties.

Expected Learning Outcomes Upon completion of the course, students should have developed 1. a solid knowledge of the main tools of basic probability 2. a general understanding of the most common mathematical tools models used in Statistics, finance, econometrics etc. 3. the skill to model simple problems and pursue their actual solution by selecting the suitable probabilistic model for problem-solving. 4. the ability to apply autonomously the learned techniques to a variety of contexts, so that it will be possible to pursue further studies or to undertake post-graduate professional training courses. 5. the ability to use the language of probability and to communicate procedures and results, by adapting the concepts used to the interlocutor in the specific case.

Probability spaces - Axioms and properties of probability Conditional probability and independence Random variables Discrete and continuous Distribution functions Expectation and Moments Moment Generating function Sums of independent random variables Convergence of random variables Weak convergence Laws of large numbers Central limit theorem Conditional expectation Introduction to Martingales

Jacod, J. & Protter, P. Probability Essentials Springer 2004 Brémaud, P. Probability Theory and Stochastic Processes Springer 2020 Lecture Notes

The whole course will be interactive and will require the students' active participation. Students will be strongly encouraged to form groups and engage in discussions so as to find themselves the solutions to the problems.

To guarantee a continuous assessment, a written midterm will be open to attending students. Homework will proposed during the course The midterm will count for 30% of the final grade. It will be then necessary to sit for a second written test, after the end of the course, aimed at assessing the second half of the course and it will therefore count for the remaining percentage of the grade (70%). In alternative, students may opt to sit for a total written exam. Non attending students will have solely the option of the final written exam (100%). Attendance requirement: at least 70% of classes. Grade refusal is not allowed. Withdrawal from assessment activities (including final exam) is allowed up until the submission of the written exam or the official communication of the oral exam result.

An interview to verify understanding and motivation.