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Probability theory


Probability theory


Academic year 2016/2017

Course ID
Teaching staff
Prof. Laura Sacerdote
Prof. Federico Polito
1st year
Teaching period
First semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Formal authority
Type of examination
Written and oral
An undergrauate level class in Probability and good knowledge of real analysis. 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 wel as  for research.


Results of learning outcomes

Students attain a detailed knowledge of the foundations of the theory of probability and related topics in measure theory. They 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.
They become able to prove new results related with the studied theory, furthermore they become used to  learn using different textbooks.


Course delivery

There will be 72 hours of lessons, including 16 hours of in class exercises. Personal training on assigned exercises is important for the success in this class.


Learning assessment methods

Final exam includes written and an oral tests. The two tests are scheduled on different dates. The written test holds until the next oral exam. Written test request the solution of two exercises and it is mandatory to pass this test to be admitted to the oral test. The oral examination includes a discussion on the written test as well as the answer to two question, taken at random by the student. Students can use textbook during the written test.

During the written test you can use books (any number you wish) but personal notes are not allowed. Peronal notes inside the books should be eliminated to use the book during the exam.



Support activities

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



Overview of elementary probability. Construction of probability measures on R and random variables. Integrals over 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 and martingale inequalities. Convergence properties of discrete time martingales. Introduction to continuous time martingales.

Suggested readings and bibliography



  • Çınlar, E., "Probability and Stochastics", Springer, 2011.

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



Final written exam a.a. 2016-17, sessions of January and February 2017:

- all the handed homeworks are well done. Students can complete the written exam by solving the exercise on Martingale.

- the evaluation of the written exam will account for:

                                the exercise on Martingale

                                the handed homeworks.

- the final evaluation will account both for the written and the oral exams. The oral will be on the whole program.

Students who have not handed homeworks should advise Prof Sacerdote in advance that they need to complete the written part consisting of exercises on the whole program. From the sessions of June all the students will complete the exam with a written and an oral exam on the whole program.

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


Last update: 03/05/2017 10:13
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