Learning objectives

The course will provide a self-contained and mathematically rigorous introduction to modern techniques of data analysis and modeling of random phenomena, with special emphasis to the theoretical bases, typical of probability theory, necessary to develop effective solutions to the challenges characterizing heterogeneous areas, eg , finance, fault-detection, innovation forecasting, energy prediction, etc., typical of Industry 4.0, with particular reference to the challenges posed in the field of big data analytics. The presentation of concepts, problems and related theoretical / practical solutions will be oriented to the applications, also making use of specific statistical software (e.g. Matlab, R, KNIME, etc.) always maintaining a high level of mathematical rigor. The course will discuss the basics of modern Probability theory (eg: random variables, their distributions and main statistical properties, convergence theorems and applications), with particular attention to the fundamental stochastic processes (eg: Markov chains , birth and death processes, code theory with real world applications) and their applications within real world scenarios characterized by the presence of big data and related time series.

At the end of the course the student has to show to have acquired the following skills:
- knowledge of the formal basis of probability theory
- ability to use the concepts of random variables (both in a discrete and continuous environment)
- ability to develop models based on known probabilistic models, e.g., v.a. binomial, Poisson, Gaussian, Gaussian mixtures, etc.
- understanding and knowing how to use the basic theory of stochastic processes, with particular reference to Markov chain theory (discrete and continuous time), birth and death processes and related applications
- know and know how to use the basic notions in descriptive and inferential statistics

Prerequisites and basic notions

Basic competencies of Mathematical Analysis and Linear Algebra. In particular, the following knowledge is expected:
-- limits; differential and integral calculus for functions of one or several variables;
-- matrices and linear systems.

Program

1. Probability, conditioning and independence.
2. Random variables and their distributions. Discrete distributions. Expectation and variance. Continuous distributions.
3. Random vectors. Independence of random variables. Covariance and correlation.
4. Limit Theorems: law of large numbers and central limit theorem. Normal approximation.
5. Normal random vectors.
6. Discrete time Markov chains. Markov Chain Monte Carlo methods.
7. Poisson processes and queuing theory. Continuous time Markov chains.

Bibliography

Didactic methods

All the topics will be illustrated in class. Additional material, such as exercises, lecture notes, and further references, will be available on Moodle page of the course.

Learning assessment procedures

The exam consists of a written test, in which students will be required to solve some exercises, followed by an oral interview during which they must demonstrate an adequate knowledge of the topics covered in the written part.
The written test can be completed by passing two midterm exams during the semester or by passing a regular sitting during an exam session.
The exam is scored out of 30 points, with a minimum passing score of 18 out of 30.
During the exam, the use of any artificial intelligence tool is strictly prohibited. Academic integrity requires students to demonstrate their own level of learning, as only through independent work and critical thinking can they acquire the skills and knowledge that will be assessed.

Evaluation criteria

The student must demonstrate that she is familiar with the basics in probability and in Markov chain theory, that she can apply theory to problem-solving and she is able to solve exercises of appropriate difficulty.

Exam language

English

Sustainable Development Goals - SDGs

This initiative contributes to the achievement of the Sustainable Development Goals of the UN Agenda 2030. More information on sustainability