OVERVIEW

The class aims at providing the basic knowledge concerning probability and it rules, discrete and continuous random variables (r.v.’s), and stochastic processes. The class contents are organized along these lines, addressing first probability basics, combinatorial analysis, discrete r.v.’s, continuous r.v.’s. (both along with the definition of 1st and 2nd moments and with examples of the main statistical distributions and probability density functions), multiple r.v.’s and their joint distribution, relevant inequalities and the Central Limit Theorem and, finally, the basics of Random Processes (stationarity, correlation and covariance functions).

AIMS AND CONTENT

LEARNING OUTCOMES

The module introduces the key concepts related to stochastic modeling in the framework of disaster risk prevention and assessment. Basic knowledge will be provided about probability theory, random variables, stochastic processes, and Bayesian decision theory. Examples of applications to problems of data modeling and analysis associated with risk applications will be discussed.

AIMS AND LEARNING OUTCOMES

The module aims to provide students with the basic concepts and analytical tools of probability theory and elementary stochastic processes. The course develops the skills required to describe uncertain phenomena through probabilistic models and to analyze the statistical properties of random variables and simple random processes.

Upon successful completion of the module, students will be able to:

• Explain the fundamental concepts of probability theory, including conditional probability, independence, and combinatorial methods.

• Apply probability rules to solve elementary probabilistic problems.

• Describe discrete and continuous random variables through probability distributions and statistical descriptors.

• Compute probabilities, expectation values, variances, and other basic statistical measures.

• Analyze joint distributions of multiple random variables and evaluate relationships through covariance and correlation.

• Apply fundamental probabilistic results, including the Central Limit Theorem and basic probabilistic inequalities.

• Describe the basic properties of random processes, including mean value, variance, and correlation functions.

• Recognize the main characteristics of stationary random processes and interpret their statistical descriptors.

• Use appropriate mathematical terminology to discuss probabilistic models and stochastic phenomena.

PREREQUISITES

None

TEACHING METHODS

The course is delivered through traditional lectures. Lectures present the theoretical foundations of probability theory, random variables, and random processes, supported by illustrative examples and applications to facilitate understanding of the mathematical concepts introduced during the course.

Classes are delivered in person and simultaneously made available via Microsoft Teams, allowing students to attend remotely when appropriate.

Students are encouraged to actively participate in class activities and to independently review the course materials. Additional information, teaching materials, announcements, and updates will be made available through the course AulaWeb page.

Additional information, teaching materials, announcements, and updates will be made available through the course AulaWeb page.

Students with valid certifications for Specific Learning Disorders (SLDs), disabilities or other educational needs are invited to contact the teacher and the School's contact person for disability at the beginning of teaching to agree on possible teaching arrangements that, while respecting the teaching objectives, take into account individual learning patterns. Contacts of the School's disability contact person can be found at the following link Comitato di Ateneo per l’inclusione delle studentesse e degli studenti con disabilità o con DSA | UniGe | Università di Genova

SYLLABUS/CONTENT

The topics covered in the lectures are:

• Probability basics
  • Basic definitions, probability rules, conditional probability, independence, combinatorial methods.
• Discrete random variables
  • Probability Mass Function, expectation, variance; discrete distributions: uniform, Bernoulli, Binomial, Poisson; multiple discrete r.v.’s.
• Continuous random variables
  • Cumulative distribution function, probability density function; expectation and variance; continuous distributions: uniform, exponential, normal; multiple continuous r.v.’s, joint distribution and density; functions of r.v.’s.
• Correlation, covariance; Markov and Chebyshev Inequalities; Central Limit Theorem.
• Random Processes.
  • Random Processes and r.v.’s; mean value and variance; autocorrelation, autocovariance, correlation coefficient; complex processes; stationarity and properties of stationary processes.

RECOMMENDED READING/BIBLIOGRAPHY

Course material available on AulaWeb (https://www.aulaweb.unige.it), including lecture slides and additional teaching resources, is sufficient for preparing for the examination.

Suggested textbook for further study:

- Dimitri P. Bertsekas and John Tsitsiklis, Introduction to Probability, 2nd Ed., Athena Scientific, 

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://courses.unige.it/10553/p/students-timetable

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The examination consists of a mandatory written test covering all topics included in the syllabus. The written test requires students to solve problems and exercises related to probability theory, random variables, probability distributions, and the basic properties of random processes. The purpose of the written examination is to assess the student's ability to apply the theoretical concepts and analytical methods presented during the course.

Students who successfully pass the written examination may optionally take an oral examination. During the oral examination, students may be asked to discuss the theoretical foundations underlying the solutions provided in the written test and to explain key concepts and methods covered in the course.

The oral examination allows students to demonstrate a deeper understanding of the course topics and may lead to either an increase or a decrease of the final grade, depending on the level of knowledge, understanding, and reasoning skills demonstrated during the discussion. The final grade is determined after the completion of the examination process.

ASSESSMENT METHODS

The written examination is intended to assess the student's achievement of the module's learning outcomes through the solution of problems and exercises covering the topics included in the syllabus. Students are expected to demonstrate their ability to apply probability rules, analyze discrete and continuous random variables, compute statistical descriptors, evaluate joint distributions, and solve elementary problems involving random processes.

Particular attention is given to the correctness of the mathematical procedures adopted, the ability to select and apply appropriate probabilistic methods, and the accuracy of the results obtained.

The optional oral examination is intended to further assess the student's understanding of the theoretical concepts underlying the methods used in the written examination. Students may be asked to explain definitions, properties, assumptions, and results related to probability theory, random variables, and random processes, as well as discuss the reasoning used to solve specific problems.

Particular attention is given to the clarity of exposition, the use of appropriate mathematical terminology, and the ability to establish connections among the different topics covered in the course.

Students with learning disorders ("disturbi specifici di apprendimento", DSA) will be allowed to use specific modalities and supports that will be determined on a case-by-case basis in agreement with the delegate of the Engineering courses in the Committee for the Inclusion of Students with Disabilities.