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Probability and Mathematical Statistics 1

A.Y. 2019/2020

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 statistics, stochastic processes and stochastic calculus.

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.

4) In the laboratory, fundamental notions will be given on generation of random numbers with preassigned distributions and the most basic applications of these techniques will be presented.

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 statistics, stochastic processes and stochastic calculus.

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.

4) In the laboratory, fundamental notions will be given on generation of random numbers with preassigned distributions and the most basic applications of these techniques will be presented.

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

**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.

5) Laboratory.

Generation of random numbers with given laws. The method of inverse distribution function and the rejection method. Applications.

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.

5) Laboratory.

Generation of random numbers with given laws. The method of inverse distribution function and the rejection method. Applications.

**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 changes 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 which is taught in the same semester.

**Teaching methods**

Classroom lectures, exercise classes, computer lab.

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.

The following further books may 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.

V. Capasso, D. Morale. Guida allo Studio della Probabilità e della Statistica Matematica. Esculapio Editore, 2013 (the chapter on simulation).

Sheldon M. Ross. A Course in Simulation. Prentice Hall, 1990.

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.

The following further books may 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.

V. Capasso, D. Morale. Guida allo Studio della Probabilità e della Statistica Matematica. Esculapio Editore, 2013 (the chapter on simulation).

Sheldon M. Ross. A Course in Simulation. Prentice Hall, 1990.

**Assessment methods and Criteria**

The final examination consists of three parts: a written exam, an oral exam and a lab exam.

- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems related to the programme of the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal and/or on the internet website of the course.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered.

-The lab exam consists in answering a question on a topic presented during the lab and usually in writing down a short code script; it takes place at the same time as the written exam and aims at assessing the student's ability to apply her/his knowledge.

The complete final examination is passed if the final evalutation of all three parts (written, oral, lab) is positive. Final marks are given using the numerical range 0-30, and will be communicated 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, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems related to the programme of the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take two midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal and/or on the internet website of the course.

- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course, in order to evaluate her/his knowledge and comprehension of the arguments covered.

-The lab exam consists in answering a question on a topic presented during the lab and usually in writing down a short code script; it takes place at the same time as the written exam and aims at assessing the student's ability to apply her/his knowledge.

The complete final examination is passed if the final evalutation of all three parts (written, oral, lab) is positive. Final marks are given using the numerical range 0-30, and will be communicated 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: 33 hours

Laboratories: 12 hours

Lessons: 45 hours

Laboratories: 12 hours

Lessons: 45 hours

Shifts:

Professors:
Fuhrman Marco Alessandro, Maurelli Mario

Turno A

Professor:
Maurelli MarioTurno B

Professor:
Morale DanielaProfessor(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