# Course Details for A.Y. 2015/2016

#### Name:

**Probabilità e Processi Stocastici 1 / Probability e Stochastic Processes 1**

### Basic information

##### Credits:

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

##### Degree(s):

Master Degree in Mathematics 1^{st} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

To learn the fundamentals results concerning convergence of random variables, discrete time martingales and Poisson porcess

### Course Content

- Convergences of random variables and their implications.
Borel Cantelli Lemmas. Uniform integrability
- Conditional expected values, discrete time martingales. Hoeffdinf inequality, stopping times, martingale convergence Theorem, optional stopping Theorem,inequalities.
- Poisson process and its properties. Construction using exponential random variables in the one dimensional case and with uniform and Poisson random variables in the general case. Applications

### Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

- Acquiring knowledge and understanding of the basics tools of convergence of random variables, martingale theory and Poisson process
- The student should be able to apply the theory studied in solving concrete problems
- Making informed judgements and choices on the suitable models and approximations
- The student should be able to comunicate and explain the topics included in the program of the course
- The student should be able to read textbooks in probability and be able to increase its knowledge alone.

### Prerequisites and Learning Activities

Basic notions of elementary probability theory

### Assessment Methods and Criteria

Written and oral examination

### Textbooks

- G. R. Grimmett, D. Stirzaker, Probability and random processes, second edition , Oxford University Press. 1992.
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- G. R. Grimmett, D. Stirzaker, Probability and random processes. Problems and solutions , Oxford University Press. 1992.
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### 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: 07 ottobre 2015, 12:07*