Learning objectives
The course aims at providing the student<br />with a basic knowledge of probability<br />theory and random variables, with<br />applications to Engineering.
Prerequisites
<br />Geometria, Analisi A
Course unit content
<br /> Course Contents<br />---------------<br />Probability theory: concepts from set theory,<br />axioms of probability theory and their consequences.<br />Elements of combinatorics. Conditional<br />probability, total probability theorem<br />and Bayes formula. Repeated trials.<br /><br />Random variables: introduction to the<br />concept of probability density function.<br />Formal definition of probability density<br />function and the cumulative<br />distribution function. Dirac delta.<br />Continuous and discrete random variables.<br /><br />Trasformations of random variables:<br />trasformation of a single random variable and<br />fundamental theorem. Expected value and Law of <br />the Unconscious Statistician (LUS).<br />Moments and moment generating function.<br />Mixed Bayes formula and continuous version of the<br />total probability theorem. Pairs of random<br />variables and their transformations. Extensions to systems of n random<br />variables. Generalization of the LUS and conditional<br />expectation theorem for n random variables.<br />Correlation. Independence and incorrelation.<br /><br />Law of large numbers and its statistical<br />interpretation. Statistical interpretation<br />of the covariance. Correlation coefficient. Central<br />limit theorem. De Moivre-Laplace theorem.<br /><br />More information at:<br /><br />- http://www.tlc.unipr.it/ferrari/teaching.html<br /><br />- http://www.tlc.unipr.it/bononi/teach.html<br /><br /><br />Homework<br />----------------<br />Weekly assignment of homework problems to<br />the students, without formal grading.<br />Solutions available in the textbook.
Full programme
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Bibliography
Textbook<br />--------<br />A. Bononi e G. Ferrari: "Teoria della probabilità e variabili aleatorie<br />con applicazioni", McGraw-Hill-Italia, marzo 2005, <br />ISBN: 88-386-62886.<br /><br />Extra solved problems<br />---------------------<br />G. Prati: "Esercizi di teoria delle variabili casuali"<br />(collection of solved exercises).<br /><br />Further reading<br />----------------<br />A. Papoulis: "Probability, Random variables, and stochastic<br />processes", McGraw-Hill, 3rd Ed., 1991.
Teaching methods
The exam is written only. Duration: 3 hours. The grade of<br />the written exam, if not smaller than 18, is<br />registered as the final grade of the exam, <br />except for special cases, at <br />the teacher's discretion, in which an additional<br />oral exam may be requested. The grade of<br />the written exam must be registered<br />before the next exam date, and expires after that date.<br />During the written exam,<br />one is allowed to bring:<br />1) a calculator;<br />2) an A4 sheet of paper with formulas.<br />During the semester, a midterm exam (around the end of April)<br />and an endterm exam (around the middle of June) will be held. All students<br />(regardless of the immatriculation year) can partecipate.
Assessment methods and criteria
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Other information
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