First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCM0014412, A.A. 2019/20

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course First cycle degree in
SC1159, Degree course structure A.Y. 2008/09, A.Y. 2019/20
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Number of ECTS credits allocated 7.0
Type of assessment Mark
Course unit English denomination PROBABILITY THEORY
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge DAVID BARBATO MAT/06

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/06 Probability and Mathematical Statistics 7.0

Course unit organization
Period First semester
Year 3rd Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 4.0 32 68.0 No turn
Lecture 3.0 24 51.0 No turn

Start of activities 30/09/2019
End of activities 18/01/2020
Show course schedule 2019/20 Reg.2008 course timetable

Examination board
Board From To Members of the board
8 Calcolo delle Probabilita' - a.a. 2019/2020 01/10/2019 30/09/2020 BARBATO DAVID (Presidente)

Prerequisites: The course requires the knowledge of basic notions of Probability, in particular discrete probability spaces, discrete and absolutely continuous real valued random variables, Law of Large Numbers and Central Limit Theorem.
Target skills and knowledge: The aim of the course is to introduce the main aspects of modern Probability Theory with the use of Measure Theory.
Examination methods: Written and oral
Assessment criteria: The written part of the exam counts for about the 60% of the final grade, while the oral part determines the remaining 40%. The written part consists in the solution of exercises, that could range from theoretical problems to applications to concrete examples. In the oral part the emphasis is on definitions, statements and proofs.
Course unit contents: Measure and probability spaces.
Integration theory. Random variables and expectations.
Independence of sigma-fields, of random variables, of events. Borel-Cantelli Lemma. Kolmogorov 0-1 law.
Convergence of sequence of random variables.
Sum of independent random variables. Strong law of large numbers.
Characteristic functions. Levy theorem. Central Limit Theorem.
Conditional expectation and martingale.
Stopping time. Optional stopping theorem.
Planned learning activities and teaching methods: Lectures in classroom, including presentation of theoretical notions, examples and applications.
Textbooks (and optional supplementary readings)
  • Williams, David, Probability with martingalesDavid Williams. Cambridge [etc.]: Cambridge university press, 1991. Cerca nel catalogo