
Course unit
PROBABILITY THEORY
SCM0014412, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2017/18
ECTS: details
Type 
ScientificDisciplinary 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 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
Examination board
Examination board not defined
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 sigmafields, of random variables, of events. BorelCantelli Lemma. Kolmogorov 01 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.


