Civil Engineering

The program is divided into two main parts, regarding the first probability theory and the second basic statistical models with relevant applications to Engineering. More in details, we will first discuss an introductory part to the course and to its purposes, presenting basic concepts about probability and statistics (descriptive and inferential), pointing out the main links. Then, the following themes will be discussed by means of an initial intuitive approach, a formal definition and, then, the exposition of several explanatory examples. Probability - Introduction to the probability spaces: references to Set Theory and to properties of sets. - Axioms of probability, definition of disjoint events, probability of complementari events, properties.- Combinatorics: definition of uniform space of probability and its properties, simple combinations, simple ordered selections, simple permutations, ordered selections with repetition, hypergeometric law, ordered selection without repetition), binomial law. - Conditional probability and law of total probabilities. - Bayes Theorem: proof, problem of false positives and reliability of the tests. - Stochastic independence: independence of two events, independence of more events, pairwise independence. - Introduction to random variables: intuition, description and definition, law of a random variable. - Discrete random variables: definitions, properties, discrete density, binomial random variables, geometric random variables, discrete uniform random variables, hypergeometric random variables, Poisson random variables. - Discrete cumulative distribution function: definition and properties. - Discrete random vectors: definition, meaning and examples; density of a discrete random vector, joint density function of a random vector, marginal density function of a random vector. - Independence of random variables: definition, meaning and decomposition of the joint discrete function, discrete conditional density function, independence of functions of random variables. - Functions of discrete random vectors: definition, meaning, discrete density function of a random vector, independence of functions of independent random vectors. - Mean of a random variable: definition, mean of real functions of random vectors, linearity of the mean, meand of Bernoulli, binomial, Poisson, and geometric random variable. - Moments of a random variable: k-th moment and centered k-th moment, variance of a random variable, Chebyshev inequality, properties of the variance , variance of sums of random variables. - Covariance of two random variables; definition, meaning and properties; linear correlation coefficient and properties. - Continuous random variables: definition and meaning, density function of a continuous random variable, remarkable models: continuous uniform random variable over an interval, exponential random variable. - Continuous random vectors: joint and marginal density function, independence, independence conditions, uniform random vector over a plane. - Unidimensional functions of continuous random variables: meaning and method to establish the probability law of a transformed random variable, unidimensional functions of continuous random vectors, convolution formula. - Continuous bidimensional random vectors: conditional density, moments of continuous random vectors, mean of continuous random vectors, computation of moments of relevant distributions. - Gaussian and standard Gaussian random variables: definition, properties, mean, variance and moments. Gaussian random variable as the transformation of a standard Gaussian random variable, density function of the standard Gaussian random variable. - Convergence in law of sequences of random variables, central limit theorems, weak law of the large numbers. Statistics - Sampling and sampling statistics. Sampling mean, sampling variance, unbiasedness of sampling mean and variance, variance of the sampling mean. - Applications of the central limit theorem: normal approximation fo the sampling mean, sampling from a normal model, exact distribution of the mean and of the variance, chi-squared and t-Student random variables. - Likelihood based inference: likelihood function, maximal likelihood estimation, calculus methods for the maximal likelihood estimation, log-likelihoof function, non regular estimation problems, sampling moments. - Method of moments: moments system, moments estimation - Some remakes on confidence intervals and hypothesis tests.

The course requires preliminary knowledge of pivotal argument from calculus, such as limits, integrals, derivatives, and sequences.

The suggested textbook is the following - S. Ross, Probabilità e statistica per l'ingegneria e le scienze, Apogeo.

The main teaching method here used is the classroom taught lecture, aimed to exhibit to the students not only the theoretical content of the course, but also a large and consistent set of examples and case studies. The students will be asked to solve exercise (stand-alone or in groups), to develop the capability to apply the competences learnt during the course, to solve exercises problems in probability, statistics and applications to engineering.

Course attendance is not compulsory, but strongly recommended.

The exam of the course is made by a written and an oral exam, after the end of the lectures. The written exam (ca. 2/3 hours) checks the ability of the students to apply the competences acquired during the semester by solving several exercises. In particular, it will be evaluated: - the ability to find a correct solution of the exercise and the logic path followed to develop the solution of the proposed questions; - the application of the competences that the student is expected to have acquired in order to solve the classwork. - the use of a proper and technical language and formalism. A sufficient grade of the written exam is necessary for the oral exam. The oral exam verifies the capability of the student to describe properly the techniques used during the solution of the written exam.

Other books suggested: - P. Baldi, Introduzione alla probabilità con elementi di statistica, McGraw-Hill; - E. Orsingher, L. Beghin, Introduzione alla probabilità. Carocci editore; - S. Ross, Calcolo delle Probabilità, Apogeo Editore.

The main teaching method here used is the classroom taught lecture, aimed to exhibit to the students not only the theoretical content of the course, but also a large and consistent set of examples and case studies. The students will be asked to solve exercise (stand-alone or in groups), to develop the capability to apply the competences learnt during the course, to solve exercises problems in probability, statistics and applications to engineering.