### Teachings

Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
MARIA INFUSINO (Tit.)
Period
Second Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/64]  MATHEMATICS [64/00 - Ord. 2017]  PERCORSO COMUNE 8 64

### Objectives

The course aims to provide the students with the foundations of Probability theory, which is the basis for the study of random phenomena and therefore plays a fundamental role not only in mathematics but in many other scientific and social disciplines.

KNOWLEDGE AND UNDERSTANDING ABILITIES:
The student should acquire a complete knowledge of the fundamental elements of Probability Calculus and Distribution Theory, and the ability to connect and compare them. Furthermore, she/he should be able to rigorously prove the main results presented in the lectures and to provide concrete examples of the theoretical notions introduced in the course.

APPLICATION CAPABILITIES:
The student should show familiarity and autonomy in applying the methods introduced in the lectures to practical examples and exercises. In particular, the student should be able to solve concrete problems by independently combining the theoretical knowledge learned during the course as well as recognizing and selecting the suitable model, following the examples provided in the lectures.

MAKING JUDGMENTS:
The course aims to stimulate the objective assessment of the teaching by students, offering them a constant comparison between the theoretical contents proposed during the lectures and their acquisition through self-study using the recommended references. The student should therefore be able to critically analyze the notions learned during the course and to use them independently also in other contexts.

COMMUNICATION SKILLS:
The student should acquire the ability to express the fundamental concepts of Probability with an appropriate technical language and a rigorous mathematical formalism.

LEARNING ABILITY:
The students will develop a study methodology and a critical analysis that allow them to interpret and deepen the problems in Probability and Statistics arising in the continuation of their studies and of the academic/working career.

### Prerequisites

The student needs to possesses elements of Combinatorial Calculus, Elementary Algebra and Analysis (limits, derivatives and integrals) introduced in the standard courses of Algebra and Analysis of the first two years of the Bachelor's Degree in Mathematics. In compliance with the didactic regulations of the degree course in Mathematics A.A. 2021/22, the final exam of this course cannot be taken without having passed the exam in Mathematical Analysis 2.

### Contents

1) Introduction to probability and its axiomatization
•Introduction to the concept of probability (classical, frequentist and subjective definition)
•Combinatorics (fundamental principles, permutations, combinations, binomial coefficients)
•Sample space and events (logic and operations on events, sequences of events and their limit)
• Algebras of sets and sigma-algebras
•Kolmogorov axiomatic definition of probability and its first consequences (almost certain events and almost impossible events)
•Elementary properties of probability, total probability and compound probability theorems
•Independence, conditional probability, Bayes' theorem

2) Random variables
•Definition of random variable and its law
•Distribution function
2.1) Discrete random variables
•Probability density in the discrete case.
•Models of discrete random variables: uniform, Bernoulli, binomial, geometric, hypergeometric, Poisson.
2.2) Absolutely continuous random variables
•Probability density in the absolutely continuous case.
•Models of absolutely continuous random variables: uniform, normal, exponential, Gamma, Beta.
2.3) Random vectors
•Discrete random vectors
•Absolutely continuous random vectors
•Mixed random vectors
•Independence of random variables
•Joint laws and marginal laws of a random vector
•Conditional distributions for discrete, continuous, mixed bivariate random vectors.

3) Mathematical expectation, variance and other indicators
•Mathematical expectation and variance of discrete and absolutely continuous random variables
•First and second moments of the most known distributions.
•Moment generating function, probability generating function, characteristic function and their properties.
•Covariance and correlation coefficient.
•Markov and Chebyshev inequality
•Other indicators (e.g. median, quantiles, mode)
•Conditional expectation

4) Operations on random variables
•Transformations of random variables and their associated density
•Moments of functions of random variables
•Transformations of bivariate random variables: sum, difference, quotient and product of two random variables
•Transformations of independent random variables
•Expectation and moments of functions of random variables
•Other notable distributions: multinomial, normal multivariate, Chi-Square, t-Student

5) Asymptotic theorems
•Convergence of random variables
•Connection between the different types of convergence
•Central limit theorem
•Weak law of large numbers and its application for approximating integrals
•Strong law of large numbers

### Teaching Methods

Compatibly with the University indications on the teaching methods depending on the evolution of the COVID-19 emergency, the lectures will be delivered in presence at the blackboard or in hybrid mode (simultaneously in presence and live online streaming). The course includes two-hour lectures (for a total of 64 hours) on the theoretical contents of the course always accompanied by illustrative examples and exercises solved together with the students. There are also additional exercise sessions (for a total of 32 hours) under the guidance of a Tutor. Some of the exercises will be carried out with the help of the statistical package R.

### Verification of learning

The evaluation will consist of a weighted average between a written and an oral exam with a final grade out of thirty. The written exam, consisting of 4 or 5 exercises similar to those carried out during the course, is scheduled in 2 hours and 30 minutes. Mastery of the discipline's tools and problem solving skills will be assessed. A student can access the oral exam only after having passed the written exam with a grade of at least 18 out of 30. During the oral exam the student will have to demonstrate that she/he has understood and assimilated the arguments developed during the lectures, presenting the notions and the proofs learned during the course with mathematical rigor. Note that an excessively insufficient response can compromise the entire oral test. A negative outcome of the oral exam requires the repetition of the entire procedure (written and oral).

Compatibly with the University indications depending on the evolution of the COVID-19 emergency, the exams will be held in person or digitally via Microsoft Teams or on an alternative platform previously agreed with the students.

### Texts

Main References
- P. Baldi, Calcolo delle probabilità, McGraw-Hill, 2011.
- G. Dall’Aglio, Calcolo delle probabilità, Zanichelli Editore, 2003.
- A. Di Crescenzo, V. Giorno, A.G. Nobile, L.M. Ricciardi, Un primo corso in probabilità, Liguori Editore, 2009.
- S. M. Ross, Calcolo delle Probabilità, Apogeo, 2007.

Other useful references:
-F. Caravenna, P. Dai Pra, Probabilità: Un'introduzione attraverso modelli e applicazioni, UNITEXT 67,Springer-Verlag, 2013.
- Neil A. Weiss Calcolo delle Probabilità, Addison Wesley, 2008.
Per un’ introduzione a R:
-S. M. Iacus, G. Masarotto, Laboratorio di statistica con R, McGraw-Hill, 2007
- G.J.Kerns, Introduction to Probability and Statistics Using R ( https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.174.1836&rep=rep1&type=pdf ), 2010.