- Docente: Andrea Mentrelli
- Credits: 6
- SSD: MAT/07
- Language: English
- Teaching Mode: Traditional lectures
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Automation Engineering (cod. 8891)
Learning outcomes
The objective of the course is to introduce advanced mathematical tools that are instrumental in many fields of automation engineering. Specific topics that are presented in the course regard fundamentals in probability theory, combinatorial calculus, random variables and calculus, stochastic processes, elements of statistics. Besides theoretical tools the course will introduce SW packages for handling stochastic variables. At the end of course students masters key statistical tools that play a role in estimation, filtering and control.
Course contents
A crash course in Python for scientific applications. Overview of the Python language: The basics and the not-so-basics. Introduction to the Python ecosystem for scientific applications, data visualization, and multithreading: Numpy, Scipy, Sympy, Matplotlib, etc. Overview of various environments for Python development, with an emphasis on IPython (which will be used throughout the course).
Introduction to the modern theory of probability. Deterministic and random experiments; sample spaces and events; the algebra of events; overview of the various approaches to the study of probability; the axioms of probability; the measure of probability.
Combinatorics. The basic principle of counting; simple permutations; simple dispositions; permutations with repetitions; dispositions with repetitions; cyclic permutations; sampling; binomial coefficients and multinomial coefficients; simple combinations; combinations with repetitions; binomial theorem; number of integer solutions of linear equations.
Conditional probability. Definitions; theorem of total probability; Bayes's formula; independent events; diagnostic tests; Bayesian filters
Random variables. Definitions of random variable; distribution function of probability; cumulative distribution function; density function; expected value; variance; skewness; kurtosis; Chebyshev's inequality.
Distributions of probability. Bernoulli distribution; binomial distribution; geometric distribution; negative binomial distribution; hypergeometric distribution; Poisson distribution; discrete uniform distribution; continuous uniform distribution; exponential distribution; Rayleigh distribution; gamma distribution; Erlang distribution;; Weilbull distribution; Gaussian distribution.
Introduction to the reliability theory. Failure rate and reliability/survival functions; mean time between failures; the role of the exponential, gamma and Weibull distributions; the bathtub curve.
Multiple random variables. Definitions; distribution function; joint and marginal probability density functions; conditional distribution functions; independent random variables; means, covariance, moments of double random variables; correlation. Linear regression and correlation analysis. Extension to the case of multiple random variables with any number of components.
Functions of random variables. Expected value and variance of the sum and product of two random variables; linear combination of random variables. Sum of discrete and continuous random variables (convolution). Sum of exponential/normal random variables. Probability density function for functions of one or more random variables. Lognormal distribution.
Limit theorems. Laws of large numbers and limit theorems; convergence of sequences of random variables; weak laws of large numbers; the central limit theorem; applications of the central limit theorem.
Markov Chains. Definition of Markov chain and transition probabilities; representation of a Markov chain by means of graph and transition matrix; absorbing and transient states; absorbing chain; the drunkard's walk; canonical form and fundamental matrix; time to absorption and absorption probabilities; ergodic and regular chains; the Ehrenfest model; limiting matrix for regular chains; fixed vector; equilibrium state; mean first passage time; mean recurrence time; reversibility.
Stochastic Processes. Definitions and fundamentals of stochastic processes, with a focus on discrete-time processes; realizations; first and second order functions; expected value and variance; autocovariance function, autocorrelation function and autocorrelation coefficient; processes with a trend; cross-correlation. White noise; random walk; counting processes; Poisson process and its properties; sum and difference of Poisson processes. Weak-sense/Strong-sense stationary processes (WSS/SSS).
Time series. Introduction to time series: approximation of mean value, variance, autocorrelation function, autocovariance function and autocorrelation coefficient; mean-square convergence and convergence in probability; weak law of large numbers; ergodic theorem.
Gaussian Variables and Gaussian Processes. Gaussian random variables; Gaussian vectors; Gaussian processes and Gaussian spaces; Gaussian white noise.
Fourier Analysis of Stochastic Processes. Fourier series; Fourier transform; Properties of the Fourier transform; discrete-time Fourier transform (DTFT); truncated DTFT; properties of the DTFT; generalized DTFT; power spectral density; Wiener-Khinchin theorem.
Readings/Bibliography
- S. M. Ross, “Introduction to probability and statistics for engineers and scientists”, 4thEdition, Academic Press
- H. Hsu, “Probability, random variables, and random processes”, 2ndEdition,Schaum's Outline Series, McGrow Hill
- A. Papoulis, S. U. Pillai, “Probability, Random Variable, and Stochastic Processes”, 4thEdition, Mc-Grow Hill
- J. VanderPlas, "A Whirlwind Tour of Python", O'Reilly
- J. VanderPlas, "Python Data Science Handbook", O'Reilly
Teaching methods
Standard lectures in which the basic theory is explained will be combined with several examples and exercises
Assessment methods
Written test and oral examination.The written test is mainly focused on exercises for the solution of which the student is expected to apply the theory learned during the course. If the evaluation of the written test is satisfactory, the student has access to the oral examination. The validity of the written test is limited to the same session of exams. The oral examination is aimed at verifying the knowledge gained by the student concerning mainly the theoretical part of class. The final mark takes into account the evaluations of both the written and the oral part of the exam.
During the course, examples of written exams (inclusive of commented solutions) are provided.
Teaching tools
Notebook or tablet PC and projector
Office hours
See the website of Andrea Mentrelli