Ateneo - UPO

Introduction to the theory of Probability and Statistical Inference

Lecture notes and solved exercises uploaded to the DIR platform

The course aims to introduce students to the theory and practice of Probability, with a focus on discrete random variables, and to the fundamental concepts of Statistics, with specific emphasis on parameter estimation

A fundamental prerequisite is a solid knowledge of single-variable differential and integral calculus. In particular, students are required to be able to compute derivatives and integrals, as well as identify maximum and minimum points of the main elementary transcendental functions and their combinations

Lectures and practical sessions

Students with disabilities, Specific Learning Disorders (SLD), or Special Educational Needs (SEN) can request dedicated services and accommodations by contacting the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti and by visiting the dedicated university web page: https://uniupo.it/it/servizi/servizi-studentesse-e-studenti-condizione-di-disabilit%C3%A0-e-dsa. After contacting the University Staff, students may contact the course instructor to discuss suitable exam arrangements and learning adaptations

The examination consists of a written exam followed by an oral exam. The written exam is designed to assess the students' ability to apply the concepts learned during the course and is structured into 2 exercises: one on Probability Theory and one on Statistics. Each exercise comprises 3 questions of increasing difficulty. Each question correctly and fully answered is awarded 5 points (for a maximum total of 30 points). The written exam is passed with a cumulative score of at least 18. During the written exam, students are permitted to consult any paper material. The oral exam includes a discussion of the written test and an examination of the underlying theory. It aims to verify the level of knowledge and understanding of the course content, as well as to evaluate the student's independent judgment and communication skills. The final grade is

- sufficient (18 - 21) if the student passes the written exam, demonstrates knowledge and understanding of the fundamental concepts of Probability Theory and Statistics, is able to identify and correct her or his own errors when appropriately guided, and presents topics in a clear and understandable manner

- fair/good (22 - 27) if the student demonstrates a broad understanding of the syllabus, is able to connect topics covered in different chapters, and displays greater autonomy in identifying and correcting any inaccuracies

- very good/excellent (28 - 30 cum laude) if the student independently masters the subject, is capable of engaging in an in-depth discussion on the most critical aspects of the program, and argues her or his conclusions with rigorous logic and precise language

- Elements of Finite Probability Theory: fundamental axioms, equiprobable outcomes and combinatorics, conditional probability, Bayes' theorem, independent events, and system reliability

- Random Variables: discrete random variables, joint random variables, independent variables, expected value, variance and covariance, binomial random variables

- Limit Theorems: Chebyshev's inequality, the law of large numbers, and the central limit theorem

- Basic Inferential Statistics: statistical samples and estimators, estimators for mean and variance, confidence intervals, parametric statistical models, and likelihood

- Shannon Entropy: entropy as a measure of uncertainty, exponential families, and parameter estimation for exponential families

In order to achieve the knowledge and skills corresponding to the minimum passing level, students are required to

- know and be able to present, both in written and oral form, the theory underlying the concepts of finite probability and discrete random variables

- be able to apply the learned theory to solve problems involving conditional probability, combinatorics, system reliability, as well as the distributions and moments of random variables

- know and be able to present, both in written and oral form, the theory underlying the concepts of statistical samples, estimators, confidence intervals for the mean and the variance, generic estimators, and parametric statistical models

- be able to apply the learned theory to solve problems that require defining parametric statistical models, computing the bias and the mean squared error for generic estimators, and determining confidence intervals for the mean and the variance

To achieve an advanced level, students are required to

- know and be able to present, both in written and oral form, the theory underlying the law of large numbers and the central limit theorem

- be able to apply the learned theory to solve problems requiring the use of Gaussian approximation

- know and be able to present, both in written and oral form, the theory underlying maximum likelihood estimation

- be able to apply the learned theory to solve problems requiring the derivation of the maximum likelihood estimator

- know and be able to present, both in written and oral form, the concept of Shannon entropy and the theory of exponential families