B7889 - Laboratorio di matematica: probabilità

Learning outcomes

Students will be able to model random experiments using probability spaces, compute probabilities via combinatorial methods, and apply conditional probability, independence, and fundamental formulas such as the chain rule, total probability, and Bayes’ theorem. They will analyze random variables and vectors—discrete and continuous—by working with distributions, expectations, variances, standard laws, and joint/marginal structures, including covariance.

Course contents

Mathematical model of a random experiment: sample space, events, probability axioms and their consequences.

○ Conditional probability and independence: chain rule, law of total probability, and Bayes’ formula.

○ Combinatorics and discrete uniform probability spaces.

○ Random variables:

Distribution (or law) and cumulative distribution function.
Discrete and (absolutely) continuous random variables: probability mass function and probability density function.
Expectation and variance.
Standard distributions: Bernoulli, binomial, Poisson, discrete uniform, continuous uniform, exponential, normal (or Gaussian).

○ Random vectors:

Joint distribution, marginal distributions, joint cumulative distribution function, independence of random variables, covariance.
Discrete random vectors: joint probability mass function and marginal distributions.

Readings/Bibliography

Lecture notes and materials provided by the instructor.

Reference textbook:
A. Pascucci, Teoria della Probabilità [https://link.springer.com/book/10.1007%2F978-88-470-4000-7 ], Springer, 2020.

Teaching methods

Lectures and problem sessions aimed at clarifying the theory as much as possible through examples.

Assessment methods

Oral exam.

Office hours

See the website of Andrea Pascucci