# Course Details

#### Name:

**Calcolo delle probabilità B / Probability B**

### Basic information

##### Credits:

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

##### Degree(s):

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

##### 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.
- Homogeneous Markov chains. Transition probality ad 1 and more steps. Equation of Chapman-Kolmogorov.
- Homogeneous Markov chains: joint distribution and finite-dimensional distributions.
- Examples: random walk with and without barriers. Birth and death processes. The Ehrenfest chain.
- Stopping time: hitting times to points and to sets. Transient and recurrent states. Absorpion probabilities.
- Number of visits to a state. Properties and caracterization of the recurrent and transient states. Decomposition of the state space.
- Stationary distribution. Stationary distribution for Birth and death processes. Reversible processes. Markov-Kakutani theorem. Irriducible chains.
- 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 good knowledge and understanding of the homogeneous Markoc chains, 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.

### Teaching Methods

**Language**: Italian

Lessons and exercises

### Assessment Methods and Criteria

Written and oral test

### Textbooks

- P. Baldi,
**Calcolo delle Probabilità**. McGraw-Hill. 2007. *testo principale *
- 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 *

### Course page updates

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

*Course information last updated on: 20 settembre 2016, 13:46*