# Fundamentals of probabilty

## Learning outcomes of the course unit

[knowledge and understanding]

Know, understand and be able to explain all the essential arguments, in particular definitions, statements of theorems and the examples explained in class.

[applying knowledge and understanding]

Be able to solve exercises and problems on the course arguments, in particular all the ""homeworks"" assigned during the lessons.

[making judgements]

Be able to check whether an object (event, sigma-field, probability, random variable, stochastic process) is well-defined and when it enjoys the properties introduced in the lectures.

[learning skills]

Be able to read and understand scientific texts which build on the knowledge of probability, random variables, discrete-time stochastic processes, convergence theorems.

## Prerequisites

Real functions of more than one variable. Pointwise and uniform limits. Apart from that the teaching is self-contained, but some knowledge of measure theory may ease the study.

## Course contents summary

This teaching covers basic aspects of modern probability theory, following Komogorov framework. Main arguments are: measure spaces, events, random variables, independence, integration, expectation, conditional expectation, discrete-time stochastic processes, martingales, uniform integrability, modes of convergence and related theorems.

## Recommended readings

Francesco Morandin - Lecture notes 2020 (developed during the course and available online after each lesson)

David Williams - Probability with Martingales

## Teaching methods

Traditional classes (48 hours). Arguments are presented in a formal way, with proofs for most statements. Much stress is given to the motivations and we include some examples of applications. There are no exercise sessions scheduled, but homework is regularly assigned during lessons and students are encouraged to do it at home and possibly ask for solutions during the teacher office hours.

## Assessment methods and criteria

The examination has both written and oral part. The written part has exercises (which require to apply definitions and properties) and theoretical problems (that require to prove something). The oral part is based on the knowledge of theory and homework.

To pass the exam the student should master the mathematical language and formalism. She must know the mathematical objects and the theoretical results of the course and she should be able to use them with ease. She should also be able to prove theorems by herself.

## Other informations

There will be an e-learning website, where the student can find video and blackboard trascriptions for each lesson, since the teaching is done through tablet PC