# Course Details for A.Y. 2015/2016

#### Name:

**Probabilità / Probability**

### Basic information

##### Credits:

*:* Bachelor Degree in Mathematics 9 CFU (b)

##### Degree(s):

Bachelor Degree in Mathematics 3^{rd} anno curriculum Generale Compulsory

##### Language:

Italian

### Course Objectives

This course provides an extensive introduction to probability theory and to its main techniques and motivations. In this course there is also an introduction to the Markov chains that are one of the most important example of random processes. On successful completion of this module, the student has the knoweledge of the basic concept in probability and knows how to apply them in the study of concrete problems.

### Course Content

- Events and their probabilities. Probability spaces. Contitional probability. Independence and relation with product spaces.
- Random variables and their distribution. Discrete and continuous random variables. Independence. Main distributions. Random vectors. Conditional distribution and conditional expectations. Functions of random variables.
- Expectation and moments. Variance and covariance. Independence. Inequalities: Cauchy-Schwarz iMarkov, and Chebyshev.
- Convergence of random variables: onvergence in law in quadratic mean and in probability. Law of large numbers.
- Generating functions. Continuity theorem. The central limit theorem.
- Markov chains. Transition probability and Chapman-Kolmogorov equations. Examples: random walk, birth-death processes. Stopping times. Classification of states. Proprieta e caratterizzazioni degli stati transienti e ricorrenti. Stationary distributions. Reversible processes. Regular chains: the Catene Markov-Kakutani theorem.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- On succesfull completion of this course, the student should have profound knowledge of basic definitions and techniques in probability, have knowledge and understanding of Markov chains, have profound knowledge of the fundamental concepts of probability theory.
- On succesfull completion of this course, the student should understand the connections with other fields of mathematics and should be aware of potential applications.
- On succesfull completion of this course, the student should make informed choice on the suitability of the probabilistic models and approximation in concrete problems.
- On succesfull completion of this course, the student should communicate the results and the understanding of its studies during the course.
- On succesfull completion of this course, the student should be have to read and understand books and seminar in elementary probability topics.

### Prerequisites and Learning Activities

Analysis and geometry at the first two years level.

### Assessment Methods and Criteria

Written and oral test

### Textbooks

- P. Baldi, Calcolo delle Probabilità , McGraw-Hill. 2007. testo addizionale
* *
- C.J. Stone, Introduction to Stochastic Processes , Waveland Press. 1972. testo addizionale
* *
- G. Grimmett D. Stirzaker, Probability and Random Processes , Oxford. 2001. libro di testo principale
* *

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 15 settembre 2015, 17:42*