# 00082 - Probability Theory

### Course Unit Page

• Teacher Patrizia Agati

• Credits 6

• SSD MAT/06

• Teaching Mode Traditional lectures

• Language Italian

## Learning outcomes

At the end of this section of the integrated course of "Statistica and Calcolo delle probabilità", the student knows the basic tools of probability calculus, with a special focus at their role in the statistical analysis. In particular, the student should be able: - to compute the probability of events, by using the axioms and the fundamental theorems of probability calculus - to identify the main discrete and continuous random variables and compute their expected values and variances - to analytically treat the random variable transforms.

## Course contents

Random experiments, event space and random events

Probability function and criteria for measauring a probability value

Kolmogorov axioms and probability theorems

Conditional probability

Bayes-Laplace theorem

Discrete random variables: probability mass function and cumulative function. Bernoulli, Binomial, Poisson and Geometric distributions

Continuous random variables: probability density function and cumulative function. Gaussian, Chi-square, t distributions

Central limit theorem

G. Cicchitelli, Statistica, Principi e Metodi, seconda edizione, Pearson Education Italia, 2012.

A. Montanari, P. Agati, D.G. Calò, Statistica con esercizi commentati e risolti, Casa Editrice Ambrosiana, Milano, 1998.

## Teaching methods

Lectures. Tutorial application.

## Assessment methods

The aim is to check that the student has learnt the basic tools of both subjects, as well as his/her critical skills in choosing the most adequate tool to solve a given problem.

The exam for the integrated course consists of a written exam and an oral exam, both compulsory.

The written exam consists of two parts, corresponding to the two units of the integrated course. It can be taken in two partial exams (each one at the end of the corresponding module) or by a total exam. The partial exam on Part II can be taken only by the students who have passed the partial exam on Part I. The part corresponding to the contents of "Calcolo delle probabilità" consists of three exercises, each one structured in several questions. Some examples of problems possibly present in this part of the exam are available in the textbook by Montanari et al., suggested in the Section "Reading/Bibliography". For information about the part of written exam corresponding to "Statistica", see Guideweb at the voice "Statistica". The written exam lasts two hours on the whole.
The oral exam consists of three questions about some topics of both units of the integrated course.
The overall evaluation is based on the outcome of both the written and the oral exam and is expressed in marks out of 30.

## Teaching tools

Blackboard; PC; videoprojector; computer laboratory
Teaching material is available at http://campus.unibo.it/ (download is allowed only to University of Bologna students)

## Office hours

See the website of Patrizia Agati