# 28032 - Applied Mathematics T

## Learning outcomes

Methodological and practical knowledge of the basic mathematical methods suitable for modelling and analysing uncertainty.

## Course contents

Requirements/Prior knowledge

A prior knowledge and understanding of basic notions of derivation and integration is required to attend with profit this course.

Fluent spoken and written Italian is a necessary pre-requisite: all lectures and tutorials, and all study material will be in Italian.

Course contents

Brief introduction to the descriptive statistics

Organization and description of the data: tables and graphs,  frequencies and  relative frequencies, histograms. Sample mean, sample median, sample variance and sample standard deviation, percentiles. Sample correlation coefficient.

Recalls of Combinatorics

Permutations with and without repetition,  combination with and without  repetition.

Probability Theory

Probability Calculus: objectives and methods. Uncertainty, events, probability evaluation. Usual evaluation methods (classical definition, relative frequency definition). The relation between probability and statistics. Events and sets. Kolmogorov axioms. Joint probability, conditional probability. Independence. Total probability theorem. Bayes formula.

Random variables: discrete and continuous random variables. Probability distribution functions. Continuous random variables with probability density function. Numerical characteristics of the random variables: expectations (means), variance, standard deviation, moments. Couples and vectors of random variables. Joint and marginal distribution functions, joint and marginal probability mass functions, joint and marginal probability density functions.  Conditional distribution functions. Independence. Numerical characteristics: means, covariance matrix, moments. Correlated and uncorrelated random variables.

Random variable models: Bernoulli scheme. Binomial, Poisson, uniform, normal, exponential random variables. Relations between some of them. Distributions coming from the normal one: the Chi-square distribution, the Student-t distribution, the Fisher distribution.

Functions of random variables: numerical characteristics: mean and variance with particular attention to some special cases (sum and products of two random variables, linear combination of an arbitrary number of random variables, the case of independent identically distributed random variables, etc.). Probability distribution function for a function of one or more than one random variables.  Monte Carlo methods.

Limit laws in probability theory: sequences of random variables and convergence. Markov and Chebyshev inequalities, weak law of large numbers, Central limit theorem.

Introduction to Random Processes: the general concept of a random process, some examples and applications. Autocorrelation function, stationary processes, wide sense stationary processes, independent increment processes, Markov chains, Markov processes, Poisson's processes.

Statistical Inference

Sampling. Confidence regions, efficiency of the estimators. Linear regression: estimate of the regression parameters, distribution of the estimators and statistical inference of the regression parameters.

S.M. ROSS, Probabilità e statistica per l'Ingegneria e le Scienze  Ed. APOGEO

Exercises and previous tests will be available in the teaching material website of the course.

## Teaching methods

The basic theory is explained in standard lessons and illustrated with several examples and exercises. Homework is also proposed every week and will be corrected during the lessons to help students in the exam preparation.

## Assessment methods

Achievements will be assessed by the means of a final exam. This is based on an analytical assessment of the "expected learning outcomes" described above. In order to properly assess such achievement the examination is composed of two different sections: written session and oral exam. The written session, duration 90 minutes, consists of 6 exercises and its possible scores are A, B, C, D or F. To be eligible to take the oral exam the student must score in the written test at least D.

The oral session, consists of a technical conversation.

Higher grades will be awarded to students who demonstrate an organic understanding of the subject, a high ability for critical application, and a clear and concise presentation of the contents.

To obtain a passing grade, students are required to at least demonstrate a knowledge of the key concepts of the subject, some ability for critical application, and a comprehensible use of technical language.

## Office hours

See the website of Francesca Brini