
Course unit
PROBABILITY THEORY
SCP4062979, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary 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 
Hours of Individual study 
Shifts 
Practice 
2.0 
26 
24.0 
No turn 
Lecture 
7.0 
56 
119.0 
No turn 
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 sigmaalgebras: generators and Borel sigmaalgebra, theorem on uniqueness of measures; independence of sigmaalgebras and BorelCantelli 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: BirkhĂ¤user, 1999.

Caravenna, Francesco; Dai Pra, Paolo, ProbabilitĂ : un'introduzione attraverso modelli e applicazioni. Milano: Springer, 2013. Testo di consultazione

Klenke, Achim, Probability theory: A comprehensive course. London: Springer, 2014. Testo di consultazione

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


