Ateneo - UPO

Introduction to the theory of Probability and Statistical Inference

Lecture notes and solved exercises uploaded to the DIR platform

The course introduces students to the theory and practice of Probability, with a focus on key standard distributions, and covers the fundamentals of Statistics, paying particular attention to 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 5 questions of increasing difficulty. Each question correctly and fully answered is awarded 3 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

- Fundamentals of probability: fundamental axioms, equally likely outcomes and combinatorics, conditional probability, Bayes' theorem, independent events, and system reliability

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

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

- Common probability distributions: Bernoulli, binomial, geometric, Pascal, exponential, Gamma, Poisson, uniform, and Gaussian random variables

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

To achieve the minimum passing grade, students are required to

- demonstrate knowledge and the ability to explain, both in written and oral form, the theory underlying the concepts of probability and random variables

- apply the acquired theoretical knowledge to solve problems involving conditional probability, combinatorics, system reliability, as well as probability distributions and moments of random variables

- demonstrate knowledge and the ability to explain, both in written and oral form, the theory regarding key standard probability distributions

- apply the acquired theoretical knowledge to solve problems involving binomial, geometric, Pascal, exponential, Gamma, Poisson, uniform, and Gaussian random variables

- demonstrate knowledge and the ability to explain, both in written and oral form, the theory underlying the concepts of statistical sample, estimators for the mean and the variance and their confidence intervals for normal samples, generic estimators, and parametric statistical models

- apply the acquired theoretical knowledge to solve problems requiring the definition of a parametric statistical model, the calculation of the bias and the mean squared error for generic estimators, and the determination of confidence intervals for the mean and the variance

To achieve an advanced grade, students are additionally required to

- demonstrate knowledge and the ability to explain, both in written and oral form, the theory underlying the law of large numbers and the central limit theorem

- apply the acquired theoretical knowledge to solve problems requiring the use of Gaussian approximation

- demonstrate knowledge and the ability to explain, both in written and oral form, the theory underlying maximum likelihood estimation

- apply the acquired theoretical knowledge to solve problems requiring the determination of the maximum likelihood estimator