## AIMS AND CONTENT

AIMS AND LEARNING OUTCOMES

The course aims to illustrate to students the general aspects of statistics and probability theory with applications to the theory of stochastic processes.

TEACHING METHODS

Online lectures according to the schedule of the Master Degree.

SYLLABUS/CONTENT

**Descriptive statistics:** populations and samples; sample mean, median and mode; Sample variance and standard deviation; sample percentiles; Chebyshev inequality; bivariate data sets and sample correlation coefficient.

**Combinatorics: **fundamental principle of combinatorics; arrangements, permutations and combinations; binomial coefficient and multinomial coefficients.

**Elements of probability:** space of outcomes and events; axioms of probability; spaces of equally likely outcomes; conditional probability; factoring of an event and Bayes formula; independent events.

**Random variables:** discrete and continuous random variables; mass and probability density functions; probability distribution function; tuples of random variables; joint distribution for discrete random variables; joint distribution for continuous random variables; independent random variables; expected value and its properties; variance and its properties; covariance and variance of the sum of random variables; moment generating function; weak law of large numbers; change of variable; sum, difference, product and quotient of random variables.

**Models of random variables:** Bernoulli and binomial random variables; Poisson's random variables; hypergeometric random variables; uniform random variables; normal random variables; exponential random variables; Gamma-type random variables; chi-square random variable; random variables of type t; F-type random variables.

**Distributions of sample statistics:** sample mean; central limit theorem; approximate distribution of the sample mean; sample variance; sample mean and variance of normal populations; sampling from finite sets.

**Parametric estimation:** maximum likelihood estimators; maximum likelihood estimator for Bernoulli variables; maximum likelihood estimator for Poisson variables; maximum likelihood estimator for normal variables; maximum likelihood estimator for uniform variables; bilateral and unilateral confidence intervals; confidence intervals for the expected value of normal distributions of known variance; confidence intervals for the expected value of distributions of unknown variance; confidence intervals for the variance of normal distributions; approximate confidence intervals for the mean of a Bernoulli distribution; confidence intervals for the mean of an exponential distribution.

Input-output systems: input-output systems; linear input-output systems; time invariant input-output systems; linear and time invariant input-output systems; convolution of functions and its properties; general notions on the Fourier transform and its properties; hints on distributions and generalized functions; frequency response of LTI I / O systems.

Stochastic processes and their spectral characteristics: stochastic process; expected value, variance and autocorrelation function of a stochastic process; cross-correlation function of two stochastic processes; stationary stochastic processes (in a weak sense); jointly stationary stochastic processes; ergodic stochastic processes; power spectrum of a stationary stochastic process and its properties; cross power spectrum of two jointly stationary stochastic processes; response of input-output systems to stochastic processes; stochastic Gaussian processes; stochastic Poisson processes.

RECOMMENDED READING/BIBLIOGRAPHY

Lecture notes.

S. Ross Introduction to Probability and Statistics for Engineers and Scientists

## TEACHERS AND EXAM BOARD

**Office hours:** Appointment on student's request (send an email to carm@sv.inge.unige.it).

Exam Board

ENRICO RIZZUTO (President)

CLAUDIO CARMELI (President Substitute)

TOMASO GAGGERO (President Substitute)

GIANVITTORIO LURIA (President Substitute)

CESARE MARIO RIZZO (President Substitute)

## LESSONS

TEACHING METHODS

Online lectures according to the schedule of the Master Degree.

LESSONS START

September 21

Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

EXAM DESCRIPTION

The exam is written and online until the end of the COVID-19 emergency. Further information about the examination will be provided on AulaWeb.

ASSESSMENT METHODS

The written examination verifies that the student has acquired and knows how to use the basic tools of probability theory (random variables, random vectors, functions of random variables, limit theorems) and statistics (estimators, hypothesis testing)

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

11/01/2021 | 14:30 | GENOVA | Scritto | |

08/02/2021 | 14:30 | GENOVA | Scritto | |

07/06/2021 | 14:30 | GENOVA | Scritto | |

05/07/2021 | 14:30 | GENOVA | Scritto | |

06/09/2021 | 14:30 | GENOVA | Scritto |