28171 - Probability II

Learning outcomes

By the end of the course the student should know the basic theory of multidimensional random variables and sequences of random variables. In particular the student should be able: - to derive the distribution of transformed random variables - to derive joint, conditional and marginal probability density functions - to state the definition and recall the properties of multivariate normal distributions – to investigate converge properties of sequences of random variables

Course contents

• Probability spaces and their elementary properties

• Various definitions of probability (just a hint)

• Random variables and their distributions

• Independence

• Probability measures on R and R^n (including distribution functions, and, in case of R, discrete, singular continuous, and absolutely continuous probability laws)

• List of the main probability measures on R and normal distribution on R^n

• Transform of random variables

• Moments

• Characteristic functions 

• Conditional distributions

• Convergence of random variables

• Laws of large numbers

• Central limit theorems

• Infinitely divisible and stable laws 

Readings/Bibliography

In order to prepare the exam, the notes are (more than) enough, provided obviously they are taken in a exhaustive and correct way. For those who don't like notes, and/or for those who want to deepen some topics, the following text books are suggested:

Bertsekas D.P. and Tsitsikli J.N., Introduction to Probability, 2nd Edition, ISBN: 978-1-886529-23-6 

Grimmett G. and Stirzaker D. (2001) Probability and random processes, Oxford University Press.

Dall'Aglio G. (1987) Calcolo delle probabilita', Zanichelli.

Teaching methods

Regular lectures and tutorials

Assessment methods

One-hour-a-half written exam, consisting of 3/4 exercises, followed by an oral examination.

The exercises of the written test are obvious versions of the ones discussed in class, and attain to elements of the syllabus covered by the course lectures. The aim of the written test is to assess the student's ability to use definitions, properties and theorems of probability theory when facing with some simple problems.

The oral part of the exam is subjected to the overcoming of the written part. The possible questions may concern each part of the course. Typically, the interview starts with a very general question (such as "Conditional distributions" or "Central limit theorem") and then, as the topic is introduced, they become more specific. In addition to knowledge of the topics discussed in the course, evaluation criteria are the skill to connect different arguments and the adequacy and consistency of the adopted language. A mnemonic exposition, as well as the inability to discuss with the teacher, are penalized. In other terms, it is important to be able to discuss with the teacher, to be interrupted, and possibly to address some simple objections.

Teaching tools

Notes and the text-books quoted above

Regarding the former, the notes of my lectures (of 2025/2026) will be available on "virtuale" within the end of the course

Office hours

See the website of Pietro Rigo