Teaching code

4S00254

Academic staff

Paolo Dai Pra, Francesca Collet

Coordinator

Paolo Dai Pra

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Period

Semester 1  dal Oct 1, 2024 al Jan 31, 2025.

Courses Single

Authorized

Lessons timetable Moodle Seminars 0

Learning objectives

The aim of the course is to present some classes of probabilistic models of particular relevance in applications, in particular dynamic models. The emphasis is placed, in addition to mathematical rigor, on developing the ability to grasp the essential aspects of a real phenomenon and translate them into a model whose analysis, analytical or numerical, is accessible.The main topic of the course is the theory of Markov chains, both in discrete and continuous time. Each development of the theory is accompanied by the presentation of examples of applicative interest, motivated by economics, physical and biological sciences, but also by computational problems that emerge in the search for efficient algorithms. In the final part of the course the notions of conditional expectation and martingale will be introduced.At the end of the course, the student will have the tools to use a wide range of probabilistic models in both theoretical and applicative contexts, understanding their limits and effective applicability, also from a computational point of view. He will also be able to have a unifying and abstract vision of classes of problems with similar characteristics, and to face the reading of advanced texts.

Prerequisites and basic notions

Basics in Probability

Program

1. Conditional expectation and conditional distribution. Martingale. Stopping theorem and convergence theorem.
2. Discrete-time Markov chains. Markov property and transition probabilities. Irreducibility, aperiodicity. Stationary distributions. Reversible distributions.
3. Hitting times. One step analysis. Convergence to the stationary distribution. Law of large numbers for Markov chains. Markov Chain Monte Carlo methods: Metropolis algorithm and Gibbs sampler.
4. Reducible Markov chains. Transient and recurrent states. Absorption probabilities.
5. Continuous-time Markov chains. The Poisson process and its properties. Continuous-time Markov property. Semigroup associated with a Markov chain: continuity and differentiability; generator. Kolmogorov equations. Stationary distributions. Dynkin's formula. Probabilistic construction of a continuous-time Markov chain.

Bibliography

Didactic methods

All the topics will be illustrated in class. Additional material, as exercises, lecture notes and further references, will be available on the Moodle page of the course.
The rights of students will be preserved in situations of travel limitation or confinement due to national provisions or in particular situations of fragile health. In these cases, you are invited to contact the teacher directly to organize the most appropriate remedial strategies.

Learning assessment procedures

The exam consists of a 180-minute written test. It includes exercises and theoretical questions, with at least one proof of those marked in the course program required.

Evaluation criteria

To pass the exam, the student must demonstrate:
-- to have understood the theoretical notions, showing detailed knowledge of definitions and statements, as well as of some proofs;
-- to be able to apply theory to problem-solving.

Criteria for the composition of the final grade

The final grade is entirely based on the outcome of the written test

Exam language

Italiano

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