Ateneo - UPO

Review of descriptive statistics on the main statistical indices.
Elements of probability: axiomatic properties of probability, conditional probability and Bayes Theorem, discrete and continuous random variables and their main models. Inferential statistics and parameter estimation. Numerical Methods: finite arithmetic and error analysis, numerical resolution of linear systems by direct methods, polynomial interpolation, numerical integration. Introduction to Octave.

PROBABILITY AND STATISTICS: + Lecture notes uploaded on the IT platform DIR. + Paolo Baldi: Calcolo delle Probabilità e Statistica, McGraw-hill, 1998. + Sheldon M. Ross: Probabilità e Statistica per l'Ingegneria e le Scienze, Apogeo Education - Seconda Edizione 2008. NUMERICAL METHODS: + Notes of the teacher uploaded on the platform DIR.

The course, comprising 3 CFU in Probability and Statistics and 3 CFU in Numerical Methods, is delivered through teaching activities (DE) consisting of traditional lectures at the blackboard. It aims to introduce students to the theory and practice of Probability Calculus, with particular emphasis on discrete random variables, as well as to the fundamental concepts of Statistics, with particular emphasis on parameter estimation. The course also aims to provide students with the fundamental knowledge of the main basic numerical methods and their properties, with particular reference to the solution of mathematical problems of interest in applied physics. The lectures will be complemented by 20 hours of practical exercises, 10 of which will take place in a computer laboratory, during which the mathematical tools introduced will also be directly applied using computers.

The course is intended to develop students’ ability to understand the meaning and underlying principles of the numerical algorithms introduced, analyse their accuracy, and assess their effectiveness in relation to the problem under consideration. Students will also acquire the ability to use appropriate mathematical language when describing numerical methods and their results.

The main educational objective is to provide students with mathematical and computational tools that are useful for addressing quantitative problems in applied physics in an informed and effective manner, while developing critical analysis skills and the ability to select the most appropriate method for a given problem.

A solid knowledge of the fundamental mathematical concepts acquired in previous courses is required. In particular, students are expected to have knowledge of functions and sequences, limits, differential calculus for functions of one variable, Taylor expansions, and integral calculus for functions of one variable. In particular, students should be able to identify the maximum and minimum points of the main elementary transcendental functions and their combinations.
Basic knowledge of linear algebra is also required, with particular reference to vector spaces, linear systems, and matrix algebra.

Teaching will consist of traditional lectures at the blackboard, aimed at presenting the theoretical concepts and numerical methods covered in the course. The lectures will be complemented by practical exercises, during which the mathematical tools introduced will also be applied using computers.

Laboratory activities will involve the active participation of students, with the aim of consolidating their understanding of the topics covered, developing practical skills in the implementation of numerical algorithms, and promoting critical analysis of the results obtained.

The concepts addressed throughout the course will be discussed and further explored collectively in class and applied during the practical exercises, with the aim of fostering independent judgment, the ability to interpret results, and the informed use of numerical methods.

Attendance: recommended.

Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/servicesstudents- physical-or-learning-disabilities Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

- Module: PROBABILITY AND STATISTICS

The examination consists of a written test followed by an oral examination.

The written test is designed to assess students’ ability to apply the concepts and methods acquired during the course and consists of two exercises, one in Probability and one in Statistics. Each exercise comprises three questions of increasing difficulty. Each question that is solved correctly and completely is awarded 5 points, for a maximum total of 30 points. The written test is considered passed if a cumulative score of at least 18 points is achieved. During the written test, students may consult any printed material.

The oral examination consists of a discussion of the written test and a further examination of the underlying theoretical concepts. It is aimed at assessing students’ level of knowledge and understanding of the course content, as well as their ability to exercise independent judgment and use appropriate technical and mathematical language.

The final grade for this module is determined according to the criteria specified in the corresponding Syllabus.

- Module: NUMERICAL METHODS

The examination consists of an oral examination covering the topics addressed in the theoretical lectures and the practical activities carried out in the computer laboratory.

During the oral examination, students are required to demonstrate that they have acquired knowledge of the course content, using appropriate mathematical tools, technical language, and terminology. The examination may include discussion of the numerical methods covered in the course, their properties, error analysis and problem conditioning, as well as aspects related to the implementation of algorithms in the Octave computing environment.

The assessment takes into account the degree of understanding of the theoretical concepts, the ability to correctly apply the numerical methods acquired, the interpretation of the results obtained, and the ability to address new problems independently.

The final grade for this module is determined according to the criteria specified in the corresponding Syllabus.

To achieve a passing grade, students must demonstrate an understanding of the fundamental concepts and the ability to correctly apply the numerical methods introduced during the course. Excellence requires, in addition to technical correctness, independent reasoning, the ability to establish connections between different topics, critically evaluate results, and contextualize numerical tools within applied problems.

Preparation for the examination is based on the lecture notes and teaching materials provided by the instructors and made available on the DIR platform, which constitute the main reference material for the topics covered in the course. The university textbooks recommended by the instructors may also be consulted to support individual study.

The examinations for the individual modules may be taken in different examination sessions, provided that both modules are passed within the same calendar year.

The final grade for the integrated course is determined jointly by the instructors of the individual modules, taking into account the grades obtained in the respective examinations.

- Module: PROBABILITY AND STATISTICS

Elements of finite probability theory: fundamental axioms, equally likely outcomes and combinatorial analysis, conditional probability, Bayes’ theorem, independent events, and system reliability.

Random variables: discrete random variables, systems of random variables, independent random 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 the mean and variance, confidence intervals, parametric statistical models, and likelihood.

Shannon entropy: entropy as a measure of uncertainty, exponential families, and estimation of their parameters.

- Module: NUMERICAL METHODS

ERROR ANALYSIS: absolute and relative error. Types of errors. Machine numbers. Representation by truncation and rounding. Machine precision. Overflow and underflow. Finite arithmetic. Problem conditioning. Conditioning of the four fundamental arithmetic operations.

SOLUTION OF LINEAR SYSTEMS: simple cases. LU factorization: existence, uniqueness, and computational cost. Pivoting. Problem conditioning. Overdetermined linear systems: least-squares solution, the normal equations method, and an introduction to methods based on QR factorization.

FUNCTION APPROXIMATION: polynomial interpolation: existence and uniqueness of the interpolating polynomial. Lagrange form. Introduction to error analysis. The discrete least-squares method.

NUMERICAL INTEGRATION: interpolatory quadrature formulas. Newton–Cotes formulas: the simple trapezoidal rule and Simpson’s rule. Error analysis. Degree of exactness of a quadrature formula. Conditioning of a definite integral and of a quadrature formula. Composite trapezoidal and Simpson’s rules and corresponding error analysis.

OCTAVE COMPUTING ENVIRONMENT: definition of scalar variables, vectors, and matrices. Variable types. Colon operator. Subvectors and submatrices. Arithmetic, relational, and logical operators. Expressions. Main built-in functions. Script M-files. Commands for data input and output. Commands for controlling the execution flow. Function M-files: input and output parameters. Main commands for 2D graphics.

At the end of the course, thanks to the teaching resources made available by the Degree Programme (lectures, practical exercises, computer laboratory activities, teaching materials, and application examples), students will have acquired fundamental knowledge and skills in Probability and Statistics and Numerical Methods. In particular, they will be able to apply the main concepts and tools of probability theory and inferential statistics, as well as understand, analyse, and use numerical methods to solve mathematical problems of interest in applied physics.

- Knowledge and understanding: study of the theoretical foundations (theorems and definitions) of statistical and numerical techniques and their applications.

Satisfactory level: students know the principles underlying the main concepts and methods of Probability and Statistics covered in the course and the main numerical methods studied. They understand the meaning of the approximations introduced and can identify the main factors affecting the accuracy and reliability of results, both in probabilistic and statistical analysis and in numerical computations.

Advanced level: students demonstrate an in-depth understanding of the concepts and methods of Probability and Statistics and of the numerical methods studied. They are able to analyse their properties, applications, advantages, and limitations, and understand the role of uncertainty, error, conditioning, and computational cost in selecting and using the most appropriate tools and algorithms.

- Applying knowledge and understanding: full ability to apply the computational and analytical techniques provided by the course. Development of the necessary software, use of available automated computational methods, and implementation of new algorithms.

Satisfactory level: students are able to apply the main concepts and methods of Probability and Statistics to familiar problems, as well as standard numerical methods to solve mathematical problems, implement simple algorithms in a computing environment, and verify the correctness of the results obtained through comparisons and error analysis.

Advanced level: students are able to apply the tools of Probability and Statistics independently and appropriately, interpreting the results obtained, as well as select the most appropriate numerical method for a given problem, justifying their choice on the basis of accuracy, stability, and computational cost. They are also able to critically analyse the results obtained, including through numerical simulations, establish connections between the different concepts and methods covered in the course, and develop solution strategies for new problems.

- Communication skills: ability to present, both orally and in writing, the details of calculations and the results obtained by applying the methods to a given problem. Ability to communicate computational procedures through a detailed analysis of the different stages of both manual calculations and automated computations implemented through software.

- Learning skills: acquisition of a good command of statistical and numerical methods, enabling students to further develop their knowledge throughout their subsequent studies.