## Studying at the University of Verona

Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.

## Academic calendar

The academic calendar shows the deadlines and scheduled events that are relevant to students, teaching and technical-administrative staff of the University. Public holidays and University closures are also indicated. The academic year normally begins on 1 October each year and ends on 30 September of the following year.

## Course calendar

The Academic Calendar sets out the degree programme lecture and exam timetables, as well as the relevant university closure dates..

Period | From | To |
---|---|---|

I semestre | Oct 1, 2013 | Jan 31, 2014 |

II semestre | Mar 3, 2014 | Jun 13, 2014 |

Session | From | To |
---|---|---|

Sessione straordinaria | Feb 3, 2014 | Feb 28, 2014 |

Sessione estiva | Jun 16, 2014 | Jul 31, 2014 |

Sessione autunnale | Sep 1, 2014 | Sep 30, 2014 |

Session | From | To |
---|---|---|

Sessione autunnale | Oct 15, 2013 | Oct 15, 2013 |

Sessione straordinaria | Dec 9, 2013 | Dec 9, 2013 |

Sessione invernale | Mar 18, 2014 | Mar 18, 2014 |

Sessione estiva | Jul 21, 2014 | Jul 21, 2014 |

Period | From | To |
---|---|---|

Vacanze Natalizie | Dec 22, 2013 | Jan 6, 2014 |

Vacanze di Pasqua | Apr 17, 2014 | Apr 22, 2014 |

Festa del S. Patrono S. Zeno | May 21, 2014 | May 21, 2014 |

Vacanze Estive | Aug 11, 2014 | Aug 15, 2014 |

## Exam calendar

Exam dates and rounds are managed by the relevant Science and Engineering Teaching and Student Services Unit.

To view all the exam sessions available, please use the Exam dashboard on ESSE3.

If you forgot your login details or have problems logging in, please contact the relevant IT HelpDesk, or check the login details recovery web page.

Should you have any doubts or questions, please check the Enrolment FAQs

## Academic staff

Squassina Marco

marco.squassina@univr.it +39 045 802 7913## Study Plan

The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University. Please select your Study Plan based on your enrolment year.

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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1° Year

Modules | Credits | TAF | SSD |
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2° Year

Modules | Credits | TAF | SSD |
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3° Year

Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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#### Legend | Type of training activity (TTA)

TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.

### Probability (2014/2015)

Teaching code

4S02753

Credits

6

Language

Italian

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

The teaching is organized as follows:

##### Teoria

##### Esercitazioni

## Learning outcomes

The course aims at introducing the Probability theory fundamentals in the more general framework of the Lebesgue measure theory. Particular attention is given to the analytical aspects of the axiomatic basis of Kolmogorov approach to modern Probability, e.g. the construction of general probability spaces, the definition of algebra, sigma-algebra, Borel sets, measurable functions, push-forward measures, etc.

The course is basically divided into two parts devoted to the definition and study of discrete, resp. continuous, random variables (r.v.).

The introduction to the fundamental concepts of the modern theory of probability is classical and based on the elements of combinatorics, the laws of set theory and on the propositional calculus fundamentals.

The approach to r.v. in the continuum is first developed in a strictly probabilistic framework, with references to some basic analytical aspects such as those of integral calculus (integration in R^n, Fubini's theorem, dominated convergence, etc.), the convolution of functions, Laplace and Fourier transforms, etc.

In a second step the probabilistic aspects are reviewed in the context of the theory of measure, especially concerning theorems of convergence for sequences of r.v., also including the central limit theorem.

During the entire course, lessons are always characterized by the presentation of examples and relevant problems. Additionally, the student are continously requested to solve exercises, of different difficulty, which are proposed by the teacher, as weel as by the tutors.

## Program

Fundamentals of Probability with respect to the axiomatic approach à la Kolmogorov

Independent/incompatible events

Rudiments of combinatorics (eg, combinations, permutations)

Uniform probability spaces

Conditional probability

Experiments with repeated independent trials

Probabilistic definition of random variable (rv)

Discrete random variables with values in R^n

o distribution function

o density function (discrete)

o Joint laws (discrete), marginals and conditional independence

o Examples: Bernoulli, binomial, geometric, Poisson, etc.

o Mean, variance and covariance operators

o Index of correlation

o Moments of a rv

o Generating Functions

Poisson approximation to the Binomial

Čebyšëv (Чебышёв) Inequality

Law of large numbers ( weak and strong formulation )

Continuous random variables with values in R^n

o Absolutely continuous rv

o Density Function (continuous)

o Joint (continuous) laws, marginals and conditional independence

o Examples: uniform, exponential, Gaussian, Gamma, etc.

o Mean, variance, covariance operators

o Normal laws

o Transformations of rv in R^n

o Conditional expectation (as a rv)

o Characteristic functions

o Moments of a rv

Convergence

o rv theory in the measure theory framework

o various types of convergence for sequences of rv

o central limit theorem and the Gaussian approximation

## Examination Methods

Written exam

## Type D and Type F activities

**Modules not yet included**

## Career prospects

## Module/Programme news

##### News for students

There you will find information, resources and services useful during your time at the University (Student’s exam record, your study plan on ESSE3, Distance Learning courses, university email account, office forms, administrative procedures, etc.). You can log into MyUnivr with your GIA login details.

## Further services

I servizi e le attività di orientamento sono pensati per fornire alle future matricole gli strumenti e le informazioni che consentano loro di compiere una scelta consapevole del corso di studi universitario.

## Graduation

## Attachments

Title | Info File |
---|---|

1. Come scrivere una tesi | 31 KB, 29/07/21 |

2. How to write a thesis | 31 KB, 29/07/21 |

4. Regolamento tesi (valido da luglio 2020) | 259 KB, 29/07/21 |

5. Regolamento tesi (valido da luglio 2022) | 171 KB, 17/02/22 |

## List of theses and work experience proposals

theses proposals | Research area |
---|---|

Formule di rappresentazione per gradienti generalizzati | Mathematics - Analysis |

Formule di rappresentazione per gradienti generalizzati | Mathematics - Mathematics |

Mathematics Bachelor and Master thesis titles | Various topics |

Stage | Research area |
---|---|

Internship proposals for students in mathematics | Various topics |

## Attendance

As stated in point 25 of the Teaching Regulations for the A.Y. 2021/2022, except for specific practical or lab activities, attendance is not mandatory. Regarding these activities, please see the web page of each module for information on the number of hours that must be attended on-site.Please refer to the Crisis Unit's latest updates for the mode of teaching.