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Probability theory


Probability theory


Academic year 2021/2022

Course ID
Teaching staff
Prof. Laura Lea Sacerdote (Lecturer)
Prof. Federico Polito (Lecturer)
Prof. Bruno Toaldo (Lecturer)
1st year
Teaching period
First semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Type of examination
Written and oral
An undergrauate level class in Probability. Good knowledge of real analysis and measure theory. Good abilities in elementary probabilistic problem solving are also necessary for the success in this class.
Propedeutic for
Stochastic Processes, Statistics for Stochastic Processes and EDS-Stochastic Dfferential Equations use concepts and tools introduced in this course.

Sommario del corso


Course objectives

Topics taught  in this class are essential tools required to a statistician and a probabilist. They are fundamental  for any modern mathematician. Students re-think to subjects  of their undergraduate studies with a different level of abstraction.This new approach allows them to control some  advanced methods of probability theory, useful for applications as well as  for research.


Results of learning outcomes

Knowledge and Understanding

Students attain a detailed knowledge of the foundations of modern theory of probability and related topics in measure theory.

Applying Knowledge and Understanding

Students attain good ability in probabilistic problem solving becoming able to deal both with theoretical and applied problems related with conditional expectation, convergence features, characteristic functions and martingales.

Making Judgements and Learning Skills

Students become able to prove new results related with the studied theory and to deepen topics by looking at suitable scientific articles and alternative textbooks.

Communication Skills

Students become able to properly use English and probability to correctly present in written and oral forms their theoretical studies and the results of exercises and homeworks.


Course delivery

The teaching course is composed of 72 hours of lectures (including 16 hours of exercises). Supporting material will be made available on Moodle in due time.

Please check this page for the teaching modalities foreseen for the a.y. 2021/22.

Some additional activities to favour direct interaction between professors and students may be organised as online meetings and/or meetings in presence, under appropriate conditions of social distancing and compatibly and in compliance with future existing regulations. For meetings in presence, students who are not able to be physically present will have the chance to follow such activities through the online teaching course material.

Webex links:



The tutoring of Friday October 1 is cancelled. Next tutoring will be Friday October 8.


Learning assessment methods

The final exam includes both a written and an oral test. The written test is valid until the following oral exam and requires the solution of two exercises and the proof of a theorem, proposition, property (selected from those discussed during lectures). It is mandatory to pass this test to be admitted to the oral test. Only for the winter session, the written test is reduced to one exercise on martingales and the proof of a theorem if the student has passed the homeworks given during the lessons. The use of textbooks and personal notes during the written test is not allowed. The oral examination includes a discussion on the written test as well as two questions. The list of the possible questions for the oral examination will be provided in advance. For the A.Y. 2021/22 the final grade is determined in the following way: let x = MAX(grade of written exam, grade of oral exam), y = MIN(grade of written exam, grade of oral exam); the final grade is given by z = 0.7*x + 0.3*y and then rounding it to the closest integer.

Students must register on the exams web-page to be admitted to the written/oral exams (in time). Unregistered students will not be admitted to exams being impossible to register their final mark.

The learning assessment method will consist of a written exam followed by an oral exam


NOTE: The written and oral exams of January 13 and January 19 will be conducted online via Webex. The webex links will be made available in due time to the students enrolled in the exams.


Support activities

The teaching course include exercises classes; extra exercises are suggested as homework.



  • Overview of elementary probability
  • Construction of probability measures on R and random variables
  • Integrals with respect to probability measures
  • Independent random variables
  • Distributions on Rn
  • Sums of random variables
  • 0-1 Laws
  • Convergence of sequences of random variables
  • Weak convergence and characteristic functions
  • Laws of large numbers and central limit theorem
  • Conditional expectations
  • Discrete time martingales
  • Optional stopping
  • Doob decomposition
  • Martingale inequalities
  • Convergence properties of discrete time martingales
  • Introduction to continuous time martingales

Suggested readings and bibliography



  • Çınlar, E., "Probability and Stochastics", Springer, 2011.
  • Durrett, R., "Probability: Theory and Examples", Cambridge, 2010.

Further suggested books: 

  • Williams, D., "Probability with Martingales", Cambridge University Press, 2001;
  • Shiryaev, A.N., "Probability", Springer, 1996;
  • Billingsey, P., "Probability and Measure", Wiley-Interscience, 1995;
  • Feller, W. "Introduction to Probability Theory and Applications", 2 Volumes, Wiley, 2008;
  • Varadhan, S.R.S., "Probability Theory", AMS, 2001;
  • Jacod, J., Protter, P., "Probability Essentials", Springer, 2004.


Class schedule


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