# Course Details for A.Y. 2015/2016

#### Name:

**Calcolo Delle Probabilità / Probability**

### Basic information

##### Credits:

*:* Laurea in Ingegneria dell'Informazione 6 CFU (a)

##### Degree(s):

Laurea in Ingegneria dell'Informazione 1^{st} anno curriculum Automatica Compulsory

##### Language:

Italian

### Course Objectives

Students should develop a certain familiarity with the basic probabilistic tools and should become able to
- model simple real problems and to propose a solution
- solve theoretical problems in secrete and continuous basic probability, using the appropriate mathematical tools.
- understand the main probabilistic structures to be able to employ them even in more complex situations;
- read a basic probability textbook;
- gain access to a more advanced probability course

### Course Content

- Probability spaces: combinatorics, axiomatic approach, uniform probability spaces,
conditional probability and independence
- Discrete random variables
Distributions: Bernoulli, binomial, Poisson, Hypergeometric, geometric,
Joint distributions and independence
Transforms of random variables, distributions of max, min, sums of random variables.
Expectation, expectation of a function of a r.v., moments, variance and covariance,
correlation, lineare regression
Examples and applications
- Continuous r.v.'s
Distribution functions: definitions and properties.
Main distributions : uniform, exponential, Gaussian, Gamma, Beta.
Distributions of transform, of max and min .
Expectation, moments, variance and covariance.
Joint distributions, sums of independent r.v.'s, conditional distributions.
Conditional expectation and conditional variance
Moment generating functions or characteristic functions
Multivariate Gaussian laws.
- Convergence of r.v.'s and approximation.
The law of large numbers, the central limit theorem, normal approximation.
- If enough time
An introduction to Markov chains: recurrence and transience, invariant probabilities,
ergodic theorem.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- Students should develop a certain familiarity with the basic probabilistic tools
- model simple real problems and to propose a solution
- solve theoretical problems in secrete and continuous basic probability,
using the appropriate mathematical tools.
- understand the main probabilistic structures to be able to employ them even in more complex
situations;
- being able to report the main points of the theory of probability and its theorems
- read a basic probability textbook;
gain access to a more advanced probability course

### Prerequisites and Learning Activities

Fundamentals of Calculus are strongly recommended

### Assessment Methods and Criteria

Written and possibly oral exams.

### Textbooks

- D. Ross, Calcolo delle probabilità , Apogeo. testo consigliato
* *
- P. Baldi, Calcolo delle probabilità , McGraw-Hill.
* *
- Schaum's outline series : Probabilità testo consigliato
* *

### 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: 22 settembre 2015, 18:46*