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Probability

A.Y. 2020/2021

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.

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.

**Lesson period:**
Second semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Second semester

PROGRAMME AND TEACHING MATERIAL

The programme and the teaching material are unchanged.

TEACHING METHODS

Lectures and exercise classes will be on line.

Lectures may be syncronous or asyncronous.

A lecture is syncronous when both teacher and attendees are on line at the same time in the official scheduled time, connected by the Microsoft Teams or Zoom platform. Lectures will be recorded (both audio and video).

Asyncronous lectures will also be given, namely audio and video recordings of the lectures will be made available in advance, and during the official timetable the teacher will review this material, adding comments, or answering questions.

The teaching method for the exercise classes will be the same.

On the website of the course, in the Ariel platform, audio and video recordings of all lectures and exercise classes will be uploaded and kept available for the whole academic year. All the related electronic material - slides, latex slides, electronic manuscripts etc. - will also be added to the website.

EXAMS

Exams will on line, with a written and an oral part.

For the written part the platform exam.net will be used, following the instructions given by the University. During the written exam students will also be connected and suitably framed by a smartphone.

During the oral part of the exam students will be connected by a computer and will be framed by its webcam. At the same time a smartphone connection will be used to frame the sheet where formulae or statements are written by the students.

MS Teams or Zoom platform will be used for the required connections, but other technical solutions are also admitted, for instance if a graphical tablet is available.

REMARK

Depending on the evolution of the sanitary situation the previous indications are subject to possibile changes. All changes will be notified on the Ariel website of the course.

The programme and the teaching material are unchanged.

TEACHING METHODS

Lectures and exercise classes will be on line.

Lectures may be syncronous or asyncronous.

A lecture is syncronous when both teacher and attendees are on line at the same time in the official scheduled time, connected by the Microsoft Teams or Zoom platform. Lectures will be recorded (both audio and video).

Asyncronous lectures will also be given, namely audio and video recordings of the lectures will be made available in advance, and during the official timetable the teacher will review this material, adding comments, or answering questions.

The teaching method for the exercise classes will be the same.

On the website of the course, in the Ariel platform, audio and video recordings of all lectures and exercise classes will be uploaded and kept available for the whole academic year. All the related electronic material - slides, latex slides, electronic manuscripts etc. - will also be added to the website.

EXAMS

Exams will on line, with a written and an oral part.

For the written part the platform exam.net will be used, following the instructions given by the University. During the written exam students will also be connected and suitably framed by a smartphone.

During the oral part of the exam students will be connected by a computer and will be framed by its webcam. At the same time a smartphone connection will be used to frame the sheet where formulae or statements are written by the students.

MS Teams or Zoom platform will be used for the required connections, but other technical solutions are also admitted, for instance if a graphical tablet is available.

REMARK

Depending on the evolution of the sanitary situation the previous indications are subject to possibile changes. All changes will be notified on the Ariel website of the course.

**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.

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.

Axiomatic definition of probability. Conditional probability and independence. Probability and random variables in sample spaces with countable outcomes. Probability on the real numbers.

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.

Attendance to each activity is not compulsory, but it is strongly recommended.

**Teaching Resources**

Textbook:

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.

P. Baldi, R. Giuliano, L. Ladelli. Laboratorio di Statistica e Probabilità, problemi svolti. McGraw Hill, 1995.

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.

P. Baldi, R. Giuliano, L. Ladelli. Laboratorio di Statistica e Probabilità, problemi svolti. McGraw Hill, 1995.

**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.

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

Lessons: 45 hours

Professors:
Fuhrman Marco Alessandro, Maurelli Mario

Educational website(s)

Professor(s)

Reception:

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.

Reception:

Monday 14-16 by appointment by email; other days by appointment by email

online meeting