PROBABILITY

Description

Objectives: 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. Assessment: 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. Teaching Methods: 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. Prerequisites: Calculus, multivariate calculus, linear algebra, elementary probability. Contents: 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 Reference Texts: Jacod, J. & Protter, P. Probability Essentials Springer 2004 Brémaud, P. Probability Theory and Stochastic Processes Springer 2020 Lecture Notes Thesis assignment criteria: An interview to verify understanding and motivation. Extended Program And Reference Reading Material: Week 1: Session 1 Probability spaces – sigma algebras Session 2 Properties of probability Session 3 Uniform probability spaces Extended Program And Reference Reading Material: Week 10: Session 1 Submartingales and supermartingales Session 2 Martingale inequalities Session 3 Exercises Extended Program And Reference Reading Material: Week 11: Session 1 Stopping times Sessione 2 Doob's optional sampling theorem Session 3 Martingale applications. Extended Program And Reference Reading Material: Week 12: If there is time: fundamentals of Markov chains. General Review and Exercises Extended Program And Reference Reading Material: Week 2: Session 1 Conditional probability Session 2 Independence Session 3 Random variables Extended Program And Reference Reading Material: Week 3: Session 1 Independence of random variables Session 2 Discrete random variables Session 3 Continuous random variables Extended Program And Reference Reading Material: Week 4: Session 1 Transformation of random variables Session 2 Joint and marginal distributions Session 3 Sum of random variables Extended Program And Reference Reading Material: Week 5: Session 1 Expectation and Moments Session 2 Covariance and correlation coefficient Session 3 Multivariate Gaussian Densities Extended Program And Reference Reading Material: Week 6: Session 1 Conditional densities and moments Session 2 Moment generating function Session 3 Types of convergence Extended Program And Reference Reading Material: Week 7: Session 1 Inequalities Session 2 Weak law of large numbers Session 3 Strong law of large numbers Extended Program And Reference Reading Material: Week 8: Session 1 Exercises Session 2 The Central Limit theorem Session 3 applications of CLT Extended Program And Reference Reading Material: Week 9: Session 1 Conditional expectation Session 2 Properties of conditional expectation session 3 Martingales Intended learning outcomes: 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.