OVERVIEW

This course introduces the mathematical foundations of probability, employing elements of abstract measure theory, and equips students with tools to rigorously model stochastic phenomena. Theoretical approaches are supplemented with examples and exercises to prepare students for applied probabilistic methods.

AIMS AND CONTENT

LEARNING OUTCOMES

The objective of the course is to provide a solid introduction to the theory of probability, utilising fundamental tools of measurement theory. The following objectives are pursued: the acquisition of a rigorous understanding of the fundamental concepts of probability, the mastery of the main limit theorems and the different notions of convergence, and the interpretation of random phenomena through probabilistic models, with links to concrete examples. Upon completion of the course, students will have the ability to formalise probabilistic problems within a mathematically rigorous framework, to prove fundamental properties of random variables, and to utilise limit theorems for the asymptotic analysis of random phenomena.

AIMS AND LEARNING OUTCOMES

By the end of the course, students will be able to:

• Master fundamental probability definitions and calculation rules.

• Understand concepts of random vectors, joint/marginal distributions, and densities.

• Compute expectations, variances, and moments.

• Distinguish between types of convergence (in law, in probability, almost sure) and their applications.

• Construct probabilistic models and derive quantities of interest.

PREREQUISITES

• Differential and integral calculus (single/multivariable functions).

• Knowledge of numerical series.

TEACHING METHODS

• Lectures: 4 hours/week (theory).

• Exercises: 3 hours/week (problem-solving).

SYLLABUS/CONTENT

1. Probability Spaces: Events, σ-algebras, probability measures, continuity properties.

2. Independence and Conditioning: Total probability, Bayes’ theorem, Borel-Cantelli lemma.

3. Random Variables: Distribution functions, expectation, variance, moments.

4. Fundamental Inequalities: Markov, Čebyšëv.

5. Key Distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric, normal, uniform, exponential, gamma, χ², Student’s *t*.

6. Random Vectors: Joint/marginal distributions, independence.

7. Characteristic Functions and applications.

8. Asymptotic Theorems:

   • Modes of convergence (in law, in probability, almost sure).

   • Normal approximation to binomial.

   • Law of large numbers.

   • Central limit theorem.

9. Conditional Expectation and properties.

RECOMMENDED READING/BIBLIOGRAPHY

• Pascucci, A. Teoria della Probabilità. Springer.

• Baldi, P. Calcolo delle Probabilità. McGraw Hill.

• Durrett, R. Elementary Probability for Applications. Cambridge.

• Durrett, R. Probability: Theory and Examples. Cambridge.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

Start Date: 23 September 2024.
Schedule: Available via EasyAcademy Portal.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The examination consists of a three-hour written test followed by an oral examination. During the written test, students are not permitted to consult notes, textbooks, or electronic devices. However, they may use a personally prepared formula sheet containing relevant formulas and results necessary for solving the exercises.

The written test is considered passed with a minimum score of 18/30; the maximum achievable score is 30 cum laude. The oral examination may be taken during the same examination session as the written test or in any subsequent session, provided it is completed within the current academic year.

Two partial written examinations are scheduled during the semester: a midterm examination at the midpoint of the course and a final examination at the end of the semester. Students will be exempt from taking the comprehensive written examination if they achieve a score of 18/30 or higher in both partial examinations​.

Should the oral examination reveal significant deficiencies in the student's preparation, the examination board reserves the right to invalidate the previously passed written test.

Students with disabilities or specific learning disorders (SLD) are referred to the "Additional Information" section for accommodation procedures.

ASSESSMENT METHODS

Examination Structure and Assessment Criteria

The written examination consists of three to four structured exercises with progressively increasing difficulty levels, covering topics addressed in theoretical lectures. These exercise types have been thoroughly examined and discussed during tutorial sessions.

The initial questions within each exercise are designed to assess:

• Comprehension of fundamental probability concepts

• Proficiency in basic computational techniques
  Correct responses to this section demonstrate minimum passing competence (sufficient for a grade of 18/30).

The subsequent questions evaluate:

• Capacity for critical and independent analysis of probability problems

• Application of advanced theoretical results
  Successful completion of this section qualifies students for maximum evaluation (30/30 cum laude).

Oral Examination Requirements:
Candidates must demonstrate:

• Clear presentation of core course concepts

• Mastery of principal theorems and their proofs

• May include supplementary exercises to verify probabilistic calculation skills

Final Grade Determination:
The assessment significantly weights the written examination results, which may be adjusted based on oral examination performance. The oral component can either:

• Elevate the written test score, or

• Reduce the originally obtained grade

FURTHER INFORMATION

Students with disabilities or specific learning disorders (SLD) who require exam accommodations must first upload the relevant certification to the university website at servizionline.unige.it, under the "Students" section. The documentation will be verified by the University’s Inclusion Services for Students with Disabilities and SLD.

Subsequently, at least 10 days before the exam date, students must send an email to the course instructor, copying:

• The School’s Disability and SLD Inclusion Coordinator (sergio.didomizio@unige.it)

• The aforementioned Inclusion Services

The email must include:

• The course name

• The exam date

• The student’s full name and ID number

• The requested compensatory tools and accommodations, specifying their functional necessity

The coordinator will confirm to the instructor that the student is eligible for exam accommodations, which must then be agreed upon with the instructor. The instructor will subsequently communicate whether the requested accommodations can be implemented.

Important deadlines:

• Requests must be submitted at least 10 days in advance to allow sufficient time for evaluation.

• For concept maps (which must be significantly more concise than study materials), late submissions may not allow enough time for necessary adjustments.

For further details on requesting accommodations, please consult the  Linee guida per la richiesta di servizi, di strumenti compensativi e/o di misure dispensative e di ausili specifici