# Course Details for A.Y. 2015/2016

#### Name:

**Calcolo delle probabilità B / Probability B**

### Basic information

##### Credits:

*:* Bachelor Degree in Mathematics 6 CFU (b)

##### Degree(s):

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

##### Language:

Italian

### Course Objectives

The student will acquire the knowledge of probability theory on general metrical spaces and will be able to apply it to the analysis of concrete problems.

### Course Content

- Measure spaces: algebras and sigma-algebras. Sigma-algebras generated by a family of subsets of the space of the outcomes. First Lemma of Borel-Cantelli. Charathèodory theorem. lebesgue measure in an interval.
- Measurable functions and random variables. Elementary properties of random variables. Distribution functions and their properties. Existence of a random variable with a given distribution function.
- Independence, definition. Second lemma of Borel Cantelli. Family of random variables independent and identically distribuited. $sigma-algebre all'infinito. Legge 0-1 di Kolmogorov.
- Absolutely continuous random variables: density function. Independence expectation and moments.
- Esponential, normal, Gamma, Beta and Cauchy distributions and thier properties.
- Random vector, definition. Random vector with distribution absolutely continuous. Multivariate normal distributions. Functions of random variables.
- Conditional probability and conditioned expected value. Definition and fundamental theorems. Case of absolutely continuous random variables.
- Generating functions and their applications.
- Limit theorems. Modes of convergence. Law of large numbers. Strong law of large numbers. Central limit theorem. Applications.

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- have profound knowledge of the fundamental concepts and techniques in probability, have knowledge and understanding of the conditional probability, have profound knowledge of the limit theorems.

On succesfull completion of this course, the student should understand the connections of probability with other fields of mathematics and should be aware of potential applications.

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 able to read and understand books and seminar in advanced probability topics.

### Prerequisites and Learning Activities

The basic courses in analisys of the first two years.
Elementary theory of the probability. Homogeneous Markov chains.

### Assessment Methods and Criteria

Written and oral test

### Textbooks

- S. Ross, Calcolo delle probabilità , Apogeo. 2004. testo di approfondimento
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- P. Baldi, Calcolo delle Probabilità , McGraw-Hill. 2007. testo di approfondimento
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- P. Billingsley, Probability and Measure , John Wiley & Sons. 1986. testo di approfondimento
* *
- G. Grimmett and D. Stirzaker , Probability and Random Processes , Oxford. 2001. testo principale
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- David Williams, Probability with martingale , Cambridge University press. 1991. 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: 16 dicembre 2015, 18:36*