OVERVIEW

This course is dedidcet to introduce basic concepts in Probability Theory. the aim is to give a solid background and the instrumentes to understand the probabilistic language: the student shound be able to build and analyse easy stochastic models. The links with other disciplines as Analysis and Statistics will be presented.

AIMS AND CONTENT

LEARNING OUTCOMES

Introduction to modeling of random phenomena.

AIMS AND LEARNING OUTCOMES

The expected learning outcomes stipulate that the student should be able to handle the basic definitions of probability spaces , the elementary rules of computation, the concept of conditioning and independence, that he/she has acquired the notion of random variable and random vector, of the distribution and possible joint and marginal density with knowledge of the role of their main characteristics (mean, variance, moments, generating functions). The student should be able to construct simple probabilistic models (possibly adapting classical schemes) in the discrete and continuous and to discuss the results given by the models.

PREREQUISITES

For this teaching, it may be useful to know how to handle basic tools of analysis, especially integral calculus and numerical series. In addition, explicit references to the basic tools of descriptive statistics will be made throughout the course.

TEACHING METHODS

Teaching involves theory (four hours per week) and exercise classes (three hours per week) coordinated with each other. Approximately two or more guided (ungraded) exercises are planned to enable the student to monitor his or her preparation in progress. Exercise sheets will be uploaded to aulaweb upon completion of each topic covered.

SYLLABUS/CONTENT

Introduction of probability: assiomatic costruction of probabiloty spaces. Concept of independence, conditional probability. Bayes Theorem. Random variables: distribution function, expectation, variance (Bernoulli, Binomiale, Geometrica, Binomiale Negativa, Ipergeometrica, Normale, Uniforme, Cauchy, Esponenziale, Gamma, Chi-Quadro, t di Student,...). Markov and Chebychev inequalities. Random vectors. Characteristic functions. Convergence definitions and theorems. Law of large numbers and Central limit theorem. Stochastic simulation.

RECOMMENDED READING/BIBLIOGRAPHY

P. Baldi, Calcolo delle Probabilità

K. L. Chung, A Course in probability Theory

J. Jacod, P. Protter, Probability Essentials

TEACHERS AND EXAM BOARD

Exam Board

EMANUELA SASSO (President)

ERNESTO DE VITO

VERONICA UMANITA' (President Substitute)

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and an oral exam.

ASSESSMENT METHODS

In the written test the student is asked to solve exercises covering the entire program. The duration of the test is three hours. Students are not allowed to consult books or notes, but are advised to prepare a "formulary" with formulas and results useful for conducting the test. To participate in the written test, it is necessary to register on the UNIGE website. The written test is considered sufficient if it obtains a score greater than or equal to 18/30. Only in very exceptional cases the exam board reserves the right to lower this threshold. There are no intermediate tests that replace the written test.

The oral test is designed toverify the absence of substantial gaps in the student's preparation, so it is conducted from the deficiencies highlighted by the written test. It may be taken in the appeal of the written test or in subsequent appeals (by the end of the current academic year). In the oral test, the student is required to be able to introduce and describe the main concepts seen in class, with special attention to the statement and demonstrations of the main theorems. Exercises will also be proposed to understand whether the student is able to use the tools of the calculus of probability. If the oral examination is insufficient, highlighting fundamental deficiencies in the student's preparation, the committee reserves the right to cancel the written examination as well. 

The written and oral examination will focus mainly on the topics covered during the lectures and will aim to assess not only whether the student has achieved an adequate level of knowledge, but whether he or she has acquired the ability to critically analyze problems related to probability.

Exam schedule