Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
Second semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (64 hours)

Essential prerequisite in order to follow the course with profit is the mastery of the topics covered in the courses of Analysis 1 and 2

In particular, it will be made use of the main results on the convergence of the series, of differential calculus and integration of function in one or several variables.

Final Examination: 

There is only a final exam, which ensures the acquisition of knowledge through a written test and an oral test.

The duration of the written test will be two hours and a half, it will not be allowed to use notes or books; the test consists of several exercises (between two and four, possibly divided in multiple points) and may include theoretical questions. Admission to the oral requires a mark of at least 16/30.

The oral exam is structured as follows:
- a review of the written exam in which the teacher explains the corrections and may receive further comments or clarifications by the students, students which were not admitted to the oral are also invited to discuss the corrections, after the discussion the teacher may decide to adjust the mark of the written test;
- an oral examination in order to check the knowledge of the notions presented in class, the ability to state and prove theorems, as well as the ability to solve theoretical and/or practical problems similar to the ones addressed in the course.
The assessment of the exam will take especially into account the following parameters: argumentative rigor and originality; quality of the exposition and competence in the use of specialized vocabulary.

It is planned to assign to the oral test at most 10 points in positive or negative.

Voto Finale

The course is intended to give an introduction to the theory of probability which is at the foundation of disciplines such as statistics and of the study of stochastic processes. A secondary goal of the course is to delineate the context of application of the probabilistic approach and of random variables, and to provide the necessary information to be able to choose the correct probabilistic methods and the best random variables to describe a given phenomenon.

More specifically, the aim of the curse is:
1) to show that random phenomena can be understood by means of mathematical models based on the notions of probability space and random variable;
2) to introduce the main features of a random variable, such as, the cumulative distribution function, the discrete and continuous density function, the moments and their generating function, and to train the students to the calculus with random variables;
3) to present several notable discrete and continuous probability distributions;
4) to introduce the most relevant results on the convergence of random variables, such as the law of large numbers and the central limit theorem, and to discuss their applications to theoretical and practical problems.

By the end of the course the students are expected to:
1) be able to engage in theoretical and applied probability problems;
2) know several techniques to compute probabilities, mean values, variance, mean values of functions, and generating functions of the moments of random variables;
3) recognize and handle several notable random variables, both discrete and continuous;
4) be able to use the results on the convergence of random variables to solve theoretical and applied problems.

- Discrete probability spaces.
Discrete probabilistic models. Combinatorial calculus. Conditional probability. Bayes formula. Independent events.

- Discrete random variables.
Definition. Discrete distribution and density. Independent random variables. Expected value. Moments, variance and covariance. Markov, Chebyschev, Jensen, and Cauchy-Schwarz inequalities. Sum of random variables. Cumulative distribution function. Moment generating function. Notable distributions of discrete random variables (uniform, Bernoulli, binomial, Poisson, geometric). Sum of independent Bernoulli. Sum of independent binomials. Sum of independent Poissonian.

- Continuous random variables.
s-algebras and probability spaces. Definition. Distribution, cumulative distribution function, density, expected value and moments, moment generating function. Independent random variables. Notable classes of continuous random variables (uniform, exponential, normal). N-dimensional random variables. Joint and marginal densities. Independence and density. Sum of random variables.

- Limit theorems.
The law of large numbers. The central limit theorem.

- F. Caravenna, P. Dai Pra: Probabilità. Un’introduzione attraverso modelli e applicazioni. Springer-Verlag Italia, 2013.
- P. Baldi: Calcolo delle Probabilità. Mc Graw Hill, 2011.

Frontal lectures: 36 hours, frontal exercise sessions: 42 hours.
In the lectures the theoretical notions are developed and the techniques necessary for the application of the theory to the resolution of exercises and problems also of a practical nature are described. The frontal exercises sessions are devoted to solve problems and exercises.

Office hours: by appointment (email to the teacher)