Introduction to Probability and Mathematical Statistics

Academic year 2021/2022
Lecturer Stefano Marmi, Franco Flandoli

Integrative teaching

Ciclo Di Seminari


Examination procedure

Written and oral exam


Elementary probability (classical discrete and continuous distributions, basic rules). Elements of descriptive statistics (like empirical mean and standard deviation).


Introduction to probability measures. Random variables, Probability density and distributions, Expectation and moments, Conditional probability and independence, examples of random variables. Conditional expectation, characteristic functions. Limit theorems: Laws of Large numbers, Central Limit theorem.

Introduction to Stochastic processes in discrete and continuous time, elements of the theory of Martingales, stochastic integrals and stochastic differential equations.

Introduction to information theory. Shannon entropy. Relative entropy. Mutual information. Asymptotic equipartition property. Information theory, codes, data compression and prediction. Kelly criterion. Horse races. Graphs. Random walks on graphs. Perron-Frobenius Theorem. Google's page rank algorithm. 

Review of estimation methods. ARMA processes. GARCH and Stochastic Volatility models. Vector processes, VAR (reduced form, structural form and identification issues). Kalman Filter and Smoother. Generalized Autoregressive Score-driven (GAS) models.

Bibliographical references

J. Jacod and P. Protter, Probability Essentials, Ed Springer 2004

A.N. Shiryaev, Probability, Ed Springer

Cover-Thomas: Elements of Information Theory

Mackay: Information theory, Inference and Learning Algorithms 

Shannon, Claude E. (July 1948). "A Mathematical Theory of Communication".

James D. Hamilton, Time Series Analysis, Princeton University Press 1994.

Durbin, James, and Siem Jan Koopman, Time series analysis by state space methods, Oxford university press, 2012.