( english version )

Signals theory A ( 5 CFU ) Tel. 0521 906513 - Fax. 0521 905758 E-mail. gianluigi.ferrari@ipruniv.cce. Home page. http://bario.tlc.unipr.it/people/ferrari/

Finalità

The course aims at providing the student with a basic knowledge of probability theory and stochastic variables, with applications to Engineering.

Programma

Probability theory: concepts of set theory, axioms of probability theory and consequences. Elements of combinatorics. Conditional probability, total probability theorem and Bayes formula. Repeated trials.

Stochastic variables: introduction to the concept of probability density function. Formal definition of the probability density function and its primitive, the cumulative distribution function. Dirac delta. Continuous and discrete stochastic variables.

Trasformations of stochastic variables: trasformation of a single random variable and fundamental theorem. Expected value and expectation theorem. Moments and moment generating function. Mixed Bayes formula and continuous version of the total probability theorem. Coupled stochastic variables and transformation of couples of stochastic variables. Extensions to systems of n random variables. Expectation theorem and conditional expectation theorem for n random variables. Correlation. Independence and incorrelation.

Law of large numbers and its statistical interpretation. Statistical interpretation of the covariance. Correlation coefficient. Central limit theorem. De Moivre-Laplace theorem.

Stochastic variables: introduction to the concept of probability density function. Formal definition of the probability density function and its primitive, the cumulative distribution function. Dirac delta. Continuous and discrete stochastic variables.

Trasformations of stochastic variables: trasformation of a single random variable and fundamental theorem. Expected value and expectation theorem. Moments and moment generating function. Mixed Bayes formula and continuous version of the total probability theorem. Coupled stochastic variables and transformation of couples of stochastic variables. Extensions to systems of n random variables. Expectation theorem and conditional expectation theorem for n random variables. Correlation. Independence and incorrelation.

Law of large numbers and its statistical interpretation. Statistical interpretation of the covariance. Correlation coefficient. Central limit theorem. De Moivre-Laplace theorem.

Attività d'esercitazione

Theory lectures and solution of exercises related to the topics of the lectures. Frequent assignment of exercises to the students for home solution.

Modalità d'esame

The exam is written. Duration: 3 hours.

The grade of the written exam is valid only for the exam session where it has been obtained, and it needs to be registered before the next exam date.

During the written exam it is allowed to bring:

1) a calculator;

2) one A4 paper with formulas.

During the course a midterm (around end of April) and a final (around middle of June) will be held.

All students (regardless of the immatriculation year) can partecipate.

The grade of the written exam is valid only for the exam session where it has been obtained, and it needs to be registered before the next exam date.

During the written exam it is allowed to bring:

1) a calculator;

2) one A4 paper with formulas.

During the course a midterm (around end of April) and a final (around middle of June) will be held.

All students (regardless of the immatriculation year) can partecipate.

Propedeuticità

Geometrica, Analisi AB

Testi consigliati

. Bononi: "Appunti di Teoria dei Segnali" (handouts).

G. Prati: "Esercizi di teoria delle variabili casuali" (collection of solved exercises).

G. Prati: "Esercizi di teoria delle variabili casuali" (collection of solved exercises).

Testi d'approfondimento

M. Luise, G. M. Vitetta: "Teoria dei segnali", McGraw-Hill, 1999.

A. Papoulis: "Probability, Random variables, and stochastic processes", McGraw-Hill, 3rd Ed., 1991.

A. Papoulis: "Probability, Random variables, and stochastic processes", McGraw-Hill, 3rd Ed., 1991.