A.Y. 2021/2022
Overall hours
Learning objectives
1) The course provides basic notions and methods of probability theory, based on measure theory. Students will be taught basic topics at the level required for their further study in mathematics.
2) The course aims at giving precise and systematic knowledge on real and vector random variables (laws, moments, independence, conditional laws etc.). This is required for an appropriate understanding of probabilistic models and for further study in probability, statistics, stochastic processes, stochastic calculus and their applications.
3) The most important limit theorems of the theory of probability will be carefully presented (laws of large numbers and central limit theorem). Basic applications will be given, in particular to statistics.
Expected learning outcomes
Students will acquire the basics of probability theory required to further pursue their study towards mathematics or its applications, either in mathematical statistics or in stochastic processes or in stochastic calculus and its applications.
Course syllabus and organization

Single session

Lesson period
Second semester
More specific information on the delivery modes of training activities for academic year 2021/22 will be provided over the coming months, based on the evolution of the public health situation.
Course syllabus
1) Theory of probability.
Axiomatic definition of probability. Conditional probability and independence. Probability and random variables in sample spaces with countable outcomes. Probability on the real numbers. Probability extension theorems.

2) Random variables.
Random variables and their laws. Integration with respect to a probability measure. Expectation and moments. Random vectors and multivariate laws. Independence of random variables. Examples of random variables used in basic probability models. Characteristic functions. Multivariate Gaussian laws and Gaussian samples.

3) Convergence of random variables.
Almost sure convergence, convergence in probability, convergence in mean of order p. Laws of large numbers. Weak convergence of probability measures. Characteristic functions and convergence in law. Central limit theorem. Applications to statistics.

4) Conditional expectation.
General definition of conditional expectation given a sigma-algebra. Conditional probabilities and conditional laws. Conditional laws for Gaussian vectors.
Prerequisites for admission
Some notions presented in the courses of Mathematical Analysis 1, 2 and 3 are required, in particular integration theory (including generalized integrals, multiple integrals and change of variables) and sequences of functions. Some basic results of matrix theory are also used (products, orthogonal matrices, positive definite matrices etc.), which are taught in the course Geometry 1. During the lectures, some concepts and results of measure theory are recalled and used: these topics are systematically presented in the course Mathematical Analysis 4.
Teaching methods
Classroom lectures and exercise classes.

Attendance to each activity is not compulsory, but it is strongly recommended.
Teaching Resources
J. Jacod, Ph. Protter. Probability Essentials. Springer, 2003, 2 ed.

Lecture notes of the teacher will be freely available on the internet site of the course, covering some specific topics of the lectures.

Files with exercises and texts of past exams will also be made available on the internet site of the course.

The following books may further be consulted:
A.F. Karr. Probability. Springer, 1993.
P. Baldi. Calcolo delle Probabilità. McGraw-Hill, 2007.
Assessment methods and Criteria
The final examination consists of two parts: written and oral.

During the written exam, students must solve some exercises in the format of open-ended questions, with the aim of assessing their ability to solve problems related to the programme of the course. The duration of the written exam depends on the number and complexity of the exercises, but in any case it will not exceed three hours. Instead of the written exam of the first examination session, students may pass two midterm exams. The outcomes of tests will be made available on the internet website of the course.

Admission to the oral exam requires an adequate grade in the written part. In the oral exam students will be asked to present results taught in the course, in order to evaluate her/his knowledge and comprehension of the covered topics.

The final grade depends on the assessments of both written and oral parts and will be notified immediately after the oral examination.

A very detailed description of the examination procedures will be made available on the internet site of the course, just after the beginning of the lectures.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 48 hours
Lessons: 45 hours
Monday, 10:30 am - 1:30 pm (upon appointment, possibly suppressed for academic duties)
Department of Mathematics, via Saldini 50, office 1017. On line if required by the pandemic conditions.
Monday 14-16 by appointment by email; other days by appointment by email
online meeting