Awareness of the content and meaning of the basic theoretical results relating to elementary probability. Have understood the concepts of independence, random variable, expectation and variance. Know how to consciously reproduce the main proofs of the theory. Know how to frame and solve some probabilistic modeling problems. Know how to use the concept of random vector and understand the classic limit theorems.
Course Prerequisites
Mathematical analysis and linear algebra of the first year of the course in mathematics
Teaching Methods
Theorical lectures (56h) and exercises sessions (28h). During exercises sessions the teacher will solve exercises on the arguments develpoed during the lectures.
Assessment Methods
The exam consists of two parts. The first part is written and consists of an average of 4 exercises in which the skills that the student has achieved in calculating and solving problems regarding the course topics are evaluated. The exercises will be divided into questions of varying difficulty aimed at establishing the degree of depth in the acquisition of these skills. If the student scores a grade higher than 18/30 in the written test, he or she is admitted to the oral test. In the oral part we will focus above all on verifying the level of knowledge of the notions presented during the course, the clarity with which they are presented and the student's ability to apply them. The formulation of the grade will be obtained by considering the overall breadth and depth of learning, as well as the clarity of the presentation and the skills demonstrated in problem solving. The grade is decided by the examining commission and will be obtained from the comparison, not necessarily reduced to an arithmetic mean, of the evaluation of the written part and the oral part.
Texts
Paolo Baldi (2012) Introduzione alla probabilità con elementi di statistica McGraw-Hill
Sheldon M. Ross (2023) Probabilità e statistica per l'ingegneria e le scienze. Maggioli editore
Robert Ash (2008) Basic Pobability Theory, Dover
Contents
Extended summary
1.- Definition of probability. 2.- Probability distribution of a random number 3.- Conditional probabilities and stochastic independence 4.- Distribution of a random vector and conditional distributions in some special cases 5.- Numerical characteristics of a probability distribution: expectation, variance, moments, regression, covariance, correlation 6.- Integral transformations: characteristic function, moment generating function and their application to the calculus of distinguished probability distributions, which are of interest for statistics 7.- Some remarkable inequalities and hints of limit theorems in probability theory: elementary examples of weak laws of large numbers and Lindeberg- Levy version of the central limit theorem