00082 - Probability Theory

Learning outcomes

By the end of the course, the student should be able to use the basic (elementary) tools of probability theory, having statistical applications in mind. Specifically, the student should be able to calculate the probability of (possibly complex) events, employing the basic results of probability theory, as well as simple mathematical models involving discrete and/or absolutely continuous random variables.

 

 

 

 

Course contents

Probability spaces and their elementary properties

Various definitions of probability (just a brief mention)

Random variables and their probability distributions

Independence

Probability measures on R and R^n (including distribution functions and the classification of laws as discrete, singular continuous and absolutely continuous)

List of the main probability laws on R and multivariate Gaussian law

Transformation of variables

Moments

Conditional distributions

Characteristic functions (just a brief mention)

Convergence of random variables

Laws of large numbers

Central limit theorems

 

 

 

 

Readings/Bibliography

In order to prepare the exam, the notes (taken by the student directly) are enough, obviously provided they are correct and complete. If notes are not sufficiently clear, and/or to deepen the various topics, the following text-books are suggested:

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

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

Teaching methods

Lessons and class exercises.

Assessment methods

The first part of the exam consists of a written assignment. Usually, the latter consists in 4 or 5 simple exercises which are obvious versions of those solved by the teacher.

 

The second part of the exam, subject to overcoming the first, lies in an oral interview. The possible questions may concern each part of the course. Typically, the interview starts with a very general question (such as "Distribution functions" or "The notion of independence") 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.

 

The above remarks do not depend on whether the exam is online or in presence. However, for online interviews, it is desirable (even if not mandatory) that the camera is able to frame the sheet where the student is writing.

Teaching tools

Notes and the text-books mentioned above.

Office hours

See the website of Pietro Rigo