Probability
A.Y. 2024/2025
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
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
· Introduction to Probability
Probability spaces. Properties of probability. Borel sets. Conditional probability and independence of events.
· Random variables
Definition of random variable. Law or distribution of a random variable. Real-valued random variables and cumulative distribution function. Discrete random variables. Absolutely continuous random variables.
· Expected value
Expected value of real-valued random variables. Markov and Chebyshev inequalties. Results concerning the interchange of limit with expectation. Series of random variables. Jensen and Cauchy-Schwarz inequalities. L^p spaces. Expected value of discrete and absolutely continuous random variables.
· Independence of random variables and random vectors
Independence of collections of events and of random variables. Definition of random vector. Joint law and marginal laws. Discrete and absolutely continuous random vectors. Covariance of real-valued random variables and variance matrix of random vectors.
· Characteristic function
Definition, properties and regularity of the characteristic function. Joint and marginal characteristic functions. Characteristic function and independence.
· Gaussian random vectors
Definition and characteristic function. Independence and uncorrelatedness. Absolutely continuous gaussian random vectors.
· Convergence of sequences of random variables
Definitions of convergence: almost sure, in L^p, in probability, in law. Relations between different notions of convergence. Criteria for establishing convergence in law based on the cumulative distribution function, on the probability mass function, on the probability density function, on the characteristic function (Lévy's continuity theorem).
· Limit theorems
Weak/strong law of large numbers and central limit theorem.
· Conditional law and conditional expectation
Transition kernel and conditional law. Definition of conditional expectation with respect to the value assumed by a random variable. Definition of conditional expectation with respect to a random variable and its properties.
Probability spaces. Properties of probability. Borel sets. Conditional probability and independence of events.
· Random variables
Definition of random variable. Law or distribution of a random variable. Real-valued random variables and cumulative distribution function. Discrete random variables. Absolutely continuous random variables.
· Expected value
Expected value of real-valued random variables. Markov and Chebyshev inequalties. Results concerning the interchange of limit with expectation. Series of random variables. Jensen and Cauchy-Schwarz inequalities. L^p spaces. Expected value of discrete and absolutely continuous random variables.
· Independence of random variables and random vectors
Independence of collections of events and of random variables. Definition of random vector. Joint law and marginal laws. Discrete and absolutely continuous random vectors. Covariance of real-valued random variables and variance matrix of random vectors.
· Characteristic function
Definition, properties and regularity of the characteristic function. Joint and marginal characteristic functions. Characteristic function and independence.
· Gaussian random vectors
Definition and characteristic function. Independence and uncorrelatedness. Absolutely continuous gaussian random vectors.
· Convergence of sequences of random variables
Definitions of convergence: almost sure, in L^p, in probability, in law. Relations between different notions of convergence. Criteria for establishing convergence in law based on the cumulative distribution function, on the probability mass function, on the probability density function, on the characteristic function (Lévy's continuity theorem).
· Limit theorems
Weak/strong law of large numbers and central limit theorem.
· Conditional law and conditional expectation
Transition kernel and conditional law. Definition of conditional expectation with respect to the value assumed by a random variable. Definition of conditional expectation with respect to a random variable and its properties.
Prerequisites for admission
Some notions presented in the courses Mathematical Analysis 1, 2, 3 are required, in particular integration theory (including generalized integrals, multiple integrals, change of variables) and sequences of functions. Some basic results of the courses Geometry 1 and 2 are also used. 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. Attendance is not compulsory, but strongly recommended.
Teaching Resources
Textbook:
J. Jacod, P. Protter. Probability Essentials. Springer, 2003, 2nd ed.
Lecture notes of the teacher will be freely available on the website of the course.
Files with exercises and texts of past exams will also be made available on the website of the course.
J. Jacod, P. Protter. Probability Essentials. Springer, 2003, 2nd ed.
Lecture notes of the teacher will be freely available on the website of the course.
Files with exercises and texts of past exams will also be made available on the website of the course.
Assessment methods and Criteria
The final examination consists of two parts: written and oral. Both parts must be passed in the same session.
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.
Students must pass the written exam in order to be admitted to the oral exam. In the latter students will be asked to present results taught in the course, so as to evaluate their knowledge and comprehension of the covered topics. The final grade depends on the assessments of both written and oral parts.
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.
Students must pass the written exam in order to be admitted to the oral exam. In the latter students will be asked to present results taught in the course, so as to evaluate their knowledge and comprehension of the covered topics. The final grade depends on the assessments of both written and oral parts.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 48 hours
Lessons: 45 hours
Lessons: 45 hours
Professors:
Cosso Andrea, Fuhrman Marco Alessandro
Shifts:
Professor(s)
Reception:
Upon appointment by email
Department of Mathematics, via Saldini 50, office 1027 or on Microsoft Teams
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.