# Probability and Statistics (2020/2021)

Course code
4S02843
Credits
9
Coordinator
Paolo Dai Pra
Other available courses
Teaching is organised as follows:
ELEMENTI DI STATISTICA 3 MAT/06-PROBABILITY AND STATISTICS II semestre Paolo Dai Pra
PROBABILITA' 6 MAT/06-PROBABILITY AND STATISTICS II semestre Paolo Dai Pra

#### Learning outcomes

The course introduces basic concepts of Probability Theory, with particular emphasis on its formal description starting from its axiomatization due to A. Kolmogorov. The emphasis will be placed on the formal derivation of the modern Probability Theory starting from the axiomatization of A.N. Kolmogorov. The course aims to provide rigorous probabilistic training which allows the student to master the techniques mathematics at the base of the modern theory of probabilities, and their application in computational-modeling and economic-financial fields. With the same spirit based on mathematical rigor, elements of Descriptive Statistics and Analysis of historical series will also be introduced.

#### Syllabus

The entire course will be available online. In addition, a number of the lessons/all the lessons (see the course
schedule) will be held in-class.

Discrete probability spaces. Elements of combinatorial calculus. Conditional probability and independence.
Applications: random permutations, percolation.
Discrete random variables and distributions. Independence of random variables. Expectation and inequalities. Notable classes of discrete random variables.
Applications: the law of small numbers, the binomial model in finance, the collector's problem.
Probability spaces and general random variables.
Absolutely continuous random variables. Notable classes of absolutely continuous random variables. Absolutely continuous random vectors. The Poisson process. Normal laws.
The law of large numbers. The central limit theorem and normal approximation.
Elements of stochastic simulation.
Basic notions of inferential statistics: unbiased and efficient estimators. Normal samples. Maximum likelihood estimators.

Textbook: F. Caravenna, P. Dai Pra, Probabilità. Un'introduzione attraverso modelli e applicazioni - UNITEXT - La matematica per il 3+2. Springer-Verlag, 2013.

#### Assessment methods and criteria

Written exam, with exercises and theoretical questions.

The assessment methods could change according to the academic rules

 Author Title Publisher Year ISBN Note Francesca Caravenna, Paolo Dai Pra Probabiltà - Un primo corso attraverso modelli e applicazioni (Edizione 1) Springer-Verlag 2013