B026814 - Probability and mathematics for statistics

Italian

1) Random variables, distributions and laws, computation of probability laws. Central limit theorems.
2) Poisson processes, Markov chains. Random walk.

-P. Baldi, Calcolo delle probabilita`.
-A. di Bucchianico, E. van Berkum, F. R. Nardi, Compendio di Probabilita` e Statistica.
-S.M. Ross, Calcolo delle probabilita`.

The course aims to provide to students the knowledge and understanding of concepts and results on discrete and continuous random variables to calculate probabilities required by concrete situations and to calculate the laws (or distributions) of one-dimensional and multidimensional random variables models for known models and random variables that cannot be studied using known models. The course aims to provide to students the knowledge and understanding skills of the limit theorems (with the related proof), Poisson processes and the Markov chains (based on the Monte Carlo method) with attention to capacity development to apply these results to problems that require the modeling to a concrete situation.
Particular attention is given to developing the communicative skills needed to expose the main arguments of the lessons and to solve the problems by justifying the correct statements and deductions with correct mathematical language.

Differential and integral calculation in one variable for real functions. Basic knowledge of algebra and geometry.

Lectures and discussion and correction of homework

The exam consists of a written examination with open questions of two types. The first type in which the student should state results explained during the lessons, with the aim to verifying the knowledge, the understanding and the quality of the exposition. A second type in which the questions are conceived to assess the ability of the students to apply their skills to problem modelling and solving, and to give the rigorous justification using formule and the appropriate scientific language. Moreover, the questions will be formulated to highlight whether the student is able to choose the best probabilistic models to solve the concrete situation described in the exercise.
Additionally, the students will have the opportunity to perform two partial tests that, if both successful, will allow them to access directly to the oral examination.
The student may also choose to take an oral test in order to improve the mark of the weigted mean of the written parts.

The course introduces the basic elements of stochastic processes with particular focus on the Poisson process and Markov chains and applications.
The main topics are:
1) Random variables, distributions and laws, computation of probabilistic laws. Convergence and approximation, limit theorems.
2) Stochastic processes: Poisson processes, Markov chains: stationary distribution, classification of the states into transient and recurrent. Random walks.