## Learning objectives

Understanding and capability to communicate the foundation of probability theory. Capability to solve exercises on probability theory, to use the specific functions, to recognize and employ the random variable models studied during the course.

## Course unit content

Introduction to probability theory. Continuous and discrete random variables. Functions of one or more random variables. Mean value and moments. Random vectors. Conditional functions. Applications of the mean value. Central limit theorem.

## Full programme

Introduction, set theory, probability space, events and axioms. Discrete events, combinatorics.

Conditional probability, combined experiments, chain rule, total probability and Bayes theorem, independence, repeated trials, continuous probability space, Dirac delta function.

(approx. 12 hours)

Random variables, cumulative distribution function, probability density function, examples of continuous variables (Gaussian, exponential, uniform, gamma...) and discrete (Poisson, Bernoulli, binomial, geometric...). Functions of random variables, graphical method, fundamental theorem. Mean value and properties, variance and properties, moment theorem, generating and characteristic functions. Conditional functions.

(approx. 18 hours)

Random vectors. Two random variables and joint functions. Multiple integral. Marginal functions. Extensions to vectors. Vector functions. Transformation of two random variables. Conditional functions of random variables, conditional mean.

(approx. 10 hours)

Mean value applications. Linearity, correlation and covariance. Optimal estimation. Generating function of the sum of random variables. Central limit theorem.

(approx. 8 hours)

## Bibliography

A. Bononi, G. Ferrari

"Introduzione alla Teoria della probabilità e variabili aleatorie con applicazioni all'ingegneria e alle scienze"

Soc. Esculapio, Bologna, aprile 2008.

Sheldon M. Ross

"Introduction to probability and statistics for engineers and scientists"

Elsevier, fifth edition, 2014.

Athanasios Papoulis

"Probability and Random Variables, and Stochastic Processes".

McGraw-Hill International Editions, fourth edition, 2002.

## Teaching methods

Classroom lectures and exercises.

Homework exercises in autonomy.

## Assessment methods and criteria

Written exam, supplementary oral following teacher decision.