# PROBABILISTIC METHODS FOR CIVIL AND ENVIRONMENTAL ENGINEERING

OVERVIEW

The course provides an introduction to probability theory and statistics through applications typical of civil and environmental engineering. The course is divided equally into traditional lessons and computer laboratories in which the student learns to deal with realistic problems involving quantities affected by uncertainties.

## AIMS AND CONTENT

LEARNING OUTCOMES

The course introduces the theory of probability and statistics as tools for the representation and analysis of random phenomena typical of the field of study. The mathematical bases of the discipline are defined starting from the general definitions to arrive at operative tools in order to represent and manipulate random or uncertain quantities. The discussion is supported by examples that cover the spectrum of applications foreseen in the following courses. Applications are performed on the computer using the Matlab programming environment.

AIMS AND LEARNING OUTCOMES

The course has two learning outcomes, which are complementary. The first objective is stated in the title and concerns the understanding of the fundamentals of probability and statistics.

The study starts with a general mathematical approach necessary to clearly formulate the problems dealt with. Mathematical tools able to represent and manipulate variables affected by uncertainties are developed. General criteria for making decisions in contexts where the available data are uncertain or the amount of information is limited are discussed. Following this introduction, the course deals with typical applications of civil and environmental engineering that anticipate, by placing them in the common framework of probability theory, problems faced in subsequent courses.

The second learning aim (not declared in the title, but not less important) consists in learning the IT tools necessary to implement the mathematical techniques and solve the problems treated. Basic notions on programming in the Matlab environment are provided and numerous guided exercises in the computer lab are held.

The coexistence of the two objectives mentioned is justified by two observations supported by experience: (1) it is not possible to teach probability and statistics without having the possibility to use computational tools for the practical solution of realistic problems; (2) it is not possible to learn how to program a computer without having real problems to face and solve. The ambition of the course is therefore to translate these two weaknesses into a strength.

Teaching methods

Lectures: on the blackboard using projections

Tutorials: in the computer laboratory. Students perform computer exercises in small groups, followed by the teacher. In the final part of the course, the exercises are organized in order to simulate an exam.

SYLLABUS/CONTENT

Probability theory

Fundamentals. Events and sample space; probability: fundamental definitions and theorems; conditional and compound probability;

Random variables. Probability distribution; probability function (of a discrete random variable); probability density (of a continuous random variable); expected value; statistical moments of a random variable; linear transformations of random variables; non-linear transformations of random variables;

Models of random variables. Normal, uniform, log-normal distribution; Rayleigh, Wibull.

Successions of random variables. Bernoulli sequence, binomial distribution, geometric model. Average return period.

Random occurrences, Poisson process, exponential model.

Asymptotic distributions, Gumbel model.

Representation of the probabilistic relation between two quantities. Joint probability distribution; joint density of probability; statistically independent random variables; expected value of functions of two random variables; sum of random variables; correlation and covariance; conditional distribution of probability of a random variable.

Statistics

Descriptive statistics, distribution indices, fractiles.

Estimated expected value, statistical moments, and probability density through the application of the frequentist definition of probability.

Order statistics method for estimating the probability distribution and the fractiles.

Estimation of the parameters of the distribution by means of the moment method, the method of maximum likelihood and the linear regression method in the probability paper.

Programming

First steps in Matlab. The work environment; data types; creation of numerical data; strings; array manipulation; manipulate numerical data; manipulate strings

Operators. Elementary operators, relational operators; logical operators;

Scientific calculation. Mathematical functions; constants; matrices.

Saving and running scripts and functions; use of paths; workspace, saving and retrieving data; management of files and folders

Principles of graphics. Programming and Input / Output techniques; creation and customization of diagrams; 2D diagrams; 3D diagrams; multiple diagrams.

Programming principles. Constructs if-else-elseif, for, while.

RECOMMENDED READING/BIBLIOGRAPHY

Course notes available on Aulaweb

Kottegoda, N.T., and Rosso, R. (2008). Applied Statistics for Civil and Environmental Engineers, Blackwell Publishing Ltd

## TEACHERS AND EXAM BOARD

**Ricevimento:** Tuesday afternoon from 15 onwards (I semester), from 16 onwards (II semester)
Office hours on request (appointment via email)

Exam Board

GIUSEPPE PICCARDO (President)

ANDREA FREDA

## LESSONS

Teaching methods

Lectures: on the blackboard using projections

Tutorials: in the computer laboratory. Students perform computer exercises in small groups, followed by the teacher. In the final part of the course, the exercises are organized in order to simulate an exam.

## EXAMS

Exam description

Practical test (computer) and oral exam