Probability and Statistics (2021/2022)
The teaching is organized as follows:
The aim of the course is to introduce the basic concepts of probability and mathematical statistics, adding concrete applications to mathematical rigor. Basic notions at the core of inferential Statistics will be also rigorously introduced, together with relevant case studies.The ultimate goal is to provide the student with the tools to be able to understand and apply rigorously, and in complete autonomy, the calculus of probability and statistics to various problems, suggested by both science and daily life. This includes the ability to analyze data, evaluate its qualitative properties to choose suitable models through an abstraction process, and the ability to read texts and articles.
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. Hypothesis testing. Significance and power of a test. Most powerful tests. Neyman-Pearson tests for simple and unilateral hypothesis. Tests for mean and variance of normal samples.
Textbook: Q. Berger, F. Caravenna, P. Dai Pra, Probabilità. Un primo corso attraverso esempi, modelli e applicazioni - UNITEXT - La matematica per il 3+2. Springer-Verlag, 2021 (Ed. 2).
Written exam, with exercises and theoretical questions.
The assessment methods could change according to the academic rules