# 78809 - Mathematical Methods for Automation Engineering M

### Course Unit Page

• Teacher Andrea Mentrelli

• Credits 6

• SSD MAT/07

• Teaching Mode Traditional lectures

• Language English

• Teaching Material

## 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

Introduction to the modern theory of probability. Deterministic and random experiments; sample spaces and events; the algebraof 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.

Random variables. Definitions of random variable; distribution function of probability; cumulative distribution function; density function; expected value; variance; skewness; kurtosis; Chebyshev's inequality.

Distributionsof 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; Weilbull distribution; Gaussian distribution.

Introduction to the reliability theory. Failure rate and reliability functions; mean time between failures; the role of the exponential, Gamma and Weibull functions; the “bathtube” 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. Extension to the case of multiple random variables with any number of components.

Functions of random variables: Representation of the expected value and variance with applications to some important cases (sum and product of two random variables; linear combination of an arbitrary number of random variables; independent random variables with the same probability density function). Probability density function for functions of one or more random variables with applications to the sum of two random variables.

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.

Introduction to signal processing: analog and digital signal processing; discrete-time signals: basic definitions, the discrete-time abstraction, digital frequency, elementary operators, energy and power, classes of discrete-time signals, finite- and infinite-length signals. Short review of Hilbert spaces: vector spaces, inner product, orthonormal bases. Fourier analysis: complex exponentials, discrete Fourier Transform (DFT), Discrete Fourier Series (DFS), Discrete-Time Fourier Transform (DTFT), relationships between transformations. Fourier transform properties. Computing the Transform: Fast-Fourier Transform (FFT). Time-frequency analysis. Discrete-Time filters: linear time-invariant systems, filtering in the time and in the frequency domain; ideal and realizable filters. The Z-transform. Filter design fundamentals.

Introduction to Stochastic Processes: The general concept of a stochastic process, distribution and density functions, autocorrelation function, stationary processes, wide sense stationary processes, independent increment processes, Markov chains, Markov processes, Poisson processes.Stochastic signal processing: spectral representation of stationary random processes; power spectral density, PSD of a stationary process; white noise.

## 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
• P. Prandoni, M. Vetterli, “Signal Processing for Communications”, CRC Press

## Teaching methods

Standard lectures in which the basic theory is explained will be combined with several examples and exercises

## Assessment methods

Written test (2 hours) 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.

## Teaching tools

Notebook or tablet PC and projector

## Office hours

See the website of Andrea Mentrelli