First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCP4062979, A.A. 2019/20

Information concerning the students who enrolled in A.Y. 2019/20

Information on the course unit
Degree course Second cycle degree in
SS1736, Degree course structure A.Y. 2014/15, A.Y. 2019/20
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Number of ECTS credits allocated 9.0
Type of assessment Mark
Course unit English denomination PROBABILITY THEORY
Website of the academic structure
Department of reference Department of Statistical Sciences
E-Learning website
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 MARKUS FISCHER MAT/06

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/06 Probability and Mathematical Statistics 9.0

Course unit organization
Period First semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 2.0 26 24.0 No turn
Lecture 7.0 56 119.0 No turn

Start of activities 30/09/2019
End of activities 18/01/2020
Show course schedule 2019/20 Reg.2014 course timetable

Examination board
Board From To Members of the board
6 Commissione a.a.2019/20 01/10/2019 30/09/2020 FISCHER MARKUS (Presidente)
BARBATO DAVID (Membro Effettivo)
CELANT GIORGIO (Membro Effettivo)

Prerequisites: Solid background in Real Analysis and Linear Algebra.
Examination methods: Written test (about three hours) requiring the solution of four problems, one of which of theoretical nature (proof of a known statement).
Assessment criteria: Final exam (written test).
Course unit contents: Basic notions: infinite sums, set operations, convergence in metric spaces.

Descriptive statistics and motivating examples.

Discrete probability models: discrete probability spaces, discrete densities; uniform spaces and combinatorics; conditional probability and independence of events; important discrete distributions (Bernoulli, binomial, hypergeometric, Poisson, geometric, negative binomial); discrete random variables and their laws; expected value and its properties.

General probability spaces and sigma-algebras: generators and Borel sigma-algebra, theorem on uniqueness of measures; independence of sigma-algebras and Borel-Cantelli lemma.

Construction of measures, product measure, distribution functions, Lebesgue measure.

General random variables and their laws. Real random variables, expected value and its properties; Lebesgue integral and transformation formula.

Absolutely continuous distributions, examples (uniform, Cauchy, exponential, Gamma, beta, Gaussian); transformations of random variables and their laws. Order statistics. Conditional distributions.

Convergence concepts for random variables. Weak and strong law of large numbers; applications. Characteristic functions, multivariate Gaussian distributions, and central limit theorem.
Planned learning activities and teaching methods: 82 hours of teaching, divided in 56 hours of theory lessons and 26 hours of exercise classes
Additional notes about suggested reading: Notes and other material will be made available through moodle.
Textbooks (and optional supplementary readings)
  • Resnick, Sidney I., A Probability Path. Boston: Birkhäuser, 1999. Cerca nel catalogo
  • Caravenna, Francesco; Dai Pra, Paolo, ProbabilitĂ : un'introduzione attraverso modelli e applicazioni. Milano: Springer, 2013. Testo di consultazione Cerca nel catalogo
  • Klenke, Achim, Probability theory: A comprehensive course. London: Springer, 2014. Testo di consultazione Cerca nel catalogo
  • Ross, Sheldon M.; edizione italiana a cura di Francesco Morandin, ProbabilitĂ  e statistica per l'ingegneria e le scienze. Santarcangelo di Romagna: Maggioli, 2014. Testo di consultazione Cerca nel catalogo