B5538 - PROBABILITA' 1

Learning outcomes

Upon completion of the course, the student will have acquired the fundamentals of probability theory, including probability measures, random variables, independence, (conditional) expected value, and limit theorems. They can apply this knowledge to scientific problems and real-world applications that involve modeling random phenomena and dealing with conditions of uncertainty.

Course contents

Complete information

Course outline:

Part 1. Measures and probability spaces. Discrete spaces and combinatorial calculus. Independence and conditional probability. General spaces: distributions and distribution functions.

Part 2. Random variables and integration. Expected value and independence. Characteristic function.

Part 3. Sequences of random variables. Law of large numbers. Central limit theorem. Monte Carlo method.

Part 4. Conditional expectation and conditional distribution. Wasserstein distance.

Further details are available on the course webpage.

Readings/Bibliography

A. Pascucci, “Probability theory I”, Springer 2024

Teaching methods

Lectures (5 credits) and problem sessions (2 credits).

Assessment methods

The exam consists of a written test and an oral examination.
The written test remains valid for all subsequent oral exam sessions: if the written test is taken more than once, the highest grade will be considered.
The written test consists of 4 or 5 exercises, of the same type as those assigned during the lectures.
The oral examination covers all the theoretical material and may include a brief discussion of the exercises from the written test. For the preparation of the theoretical part, the lecture notes or the notes available in the teaching material are sufficient, together with the Mathematica codes for some examples.
For students with SLD or temporary or permanent disabilities: students are advised to contact the relevant University office in good time (https://site.unibo.it/studenti-con-disabilita-e-dsa/it ). The office will propose any appropriate accommodations to the students concerned; these must in any case be submitted to the instructor for approval at least 15 days in advance. The instructor will assess their suitability also in relation to the learning objectives of the course.

Teaching tools

The following materials will be made available on Virtuale:

• lecture notes covering the course contents;

• Mathematica files with numerical examples;

• exercise sheets and solved past exam papers.

The software needed to read Mathematica files can be downloaded free of charge from

http://demonstrations.wolfram.com/download-cdf-player.html

 

Links to further information

https://1drv.ms/w/c/6522c228f5a947a1/IQDT-SDPMhiUSap7mdfTz7ZCAbfqvuDDrIG29GAQcNCzfI0?e=JsyRNm

Office hours

See the website of Andrea Pascucci