B8288 - CALCOLO DELLE PROBABILITA' II

Learning outcomes

By the end of the course, the student knows the basic theory of multidimensional random variables and convergence for sequences of random variables. In particular, the student is be able to: - derive the distribution of transformed random variables; - derive the joint, conditional and marginal probability density functions; - state the definition and recall the properties of multivariate normal distributions; - study the properties of convergence for successions of random variables.

Course contents

• Random vectors, expected value and covariance matrix
• Gaussian, Multinomial and Dirichlet random vectors
• Conditional expectation
• Introduction to stochastic processes
• Martingales
• Markov Chains
• Second order stationary processes
• Limits for sequences of random variables: Borel-Cantelli lemma, almost sure convergence, convergence in probability, convergence in mean square
• Convergence in distribution and characteristic function

Readings/Bibliography

Alberto Lanconelli, Introduzione alla Teoria della Probabilità - Seconda parte (2025)

Teaching methods

Lectures and tutorials

Assessment methods

One-hour written exam, articulated in a series of 2 exercises each with a maximum grade of 15 points. The possible award of "lode" is conditional on the mathematical rigour demonstrated in the solution of the exercises. Every exercise attains to elements of the syllabus covered during the course lectures. In case of online exam, this will be supported by the softwares Teams, Zoom and EOL (https://eol.unibo.it/)

Teaching tools

Exercises with solutions

Office hours

See the website of Alberto Lanconelli