Teaching code

4S00254

Academic staff

Paolo Dai Pra, Francesca Collet

Coordinatore

Paolo Dai Pra

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Period

Semester 1 dal Oct 3, 2022 al Jan 27, 2023.

Lessons timetable Moodle Seminars

Learning objectives

Moreover a student will be able to analyse some advanced applications of dynamical systems arising from population dynamics, mechanics and traffic flows. Eventually a student will be also able to produce proofs using the typical tools of modern dynamical systems and will be able to read and report specific books and articles on dynamical systems and related applications.

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 probability. 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 Moodle page of the course.
The rights of students will be preserved in situations of travel limitation or confinement due to national provisions to combat COVID 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.

Exam language

Italiano