Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
Primo semestre dal Oct 4, 2021 al Jan 28, 2022.
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
Probability, conditioning and independence.
Random variables and their distributions. Discrete distributions. Expectation and variance. Continuous distributions.
Random vectors. Independence of random variables. Covariance and correlation.
Limit Theorems: Law of Large Numbers and Central Limit Theorem. Normal approximation.
Normal random vectors.
Discrete time Markov Chains. Markov Chain Monte Carlo.
Poisson Processes and Queuing Theory. Continuous time Markov Chains.
Introduction to random networks.
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The exam takes place in two parts.
The first part, mandatory for all students, consists of a written test with exercises.
The second part can be carried out, at the student's choice, in one of the following ways:
- oral exam, in which the student must be able to present the concepts and models described in the course, both in the theoretical and in the applicative aspects;
- a project assigned by the teacher, which will include the writing of a code for a simulation.