28635 - Applied Mathematics T-A (L-Z)

Learning outcomes

A sound theoretical basis as well as a working knowledge of the fundamental mathematical methods aimed at coping with uncertainty in physical and other phenomena.

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

Foundations of probability theory. Events and sets. Kolmogorov's axioms. Joint probability, conditional probability, independence. The theorem of total probability and Bayes' formula and theorem.

Random variables. Discrete and continuous random variables. Cumulative distribution function. Continuous random variables with density. Moment-generating function. Numerical characteristics of random variables: expected value (mean), variance, standard deviation, moments.

Random vectors. Cumulative distribution function, joint density and marginal density. Conditional distribution laws. Independence. Numerical characteristics: expected values (means), covariance, correlation coefficient, moments. Correlated and uncorrelated random variables.

Some notable distributions. Bernoulli scheme. Binomial, geometric, Poisson, uniform, exponential and normal distributions. Relationships among some of these distributions and with other related distributions.

Functions of random variables. Numerical characteristics: representation of the expected value and of the variance, with applications to some notable cases (sum and product of two random variables, linear combination of an arbitrary number of random variables, and the case of independent and identically distributed random variables). Probability distribution of functions of one or more random variables: general principles and notable cases, such as the sum of two random variables, maximum and minimum, etc.

Limit theorems in probability. Chebyshev's inequality, Markov's inequality. Weak law of large numbers. Central limit theorem.

Introduction to statistics. Quantities that summarize data: sample mean, sample variance and standard deviation, percentiles. Bivariate data sets and sample correlation coefficient. Statistical inference. Sampling. Point estimators and their properties. Confidence intervals. Hypothesis testing. Linear regression.

Readings/Bibliography

Recommended textbook:

H. Hsu, Probabilità, variabili casuali e processi stocastici, ed. McGraw-Hill Italia.

Alternative textbooks:

P. Erto, Probabilità e statistica per le scienze e l'ingegneria 2/ed, ed. McGraw-Hill Italia.

A. M. Mood, F. A. Graybill, D. C. Boes, Introduzione alla statistica, ed. McGraw-Hill Italia.

 

Lecture notes and examples of exam papers from previous years will be uploaded to the Virtuale platform.

Teaching methods

The course will be based essentially on lectures delivered by the teacher and will be complemented by classroom exercises. During the lectures, some methods for modeling the analysis of uncertainty will be presented, and the mathematical tools necessary to describe and use them will be provided. Ample space will be devoted to examples and applications. In addition, students will periodically be assigned problems to solve, so that they can concretely apply the techniques presented during the lectures.

Assessment methods

The assessment is a written examination. It consists of a series of questions and exercises aimed at assessing the student’s theoretical knowledge of the basic mathematical methods for modeling and analyzing uncertainty, as well as their ability to solve problems similar to those addressed during the course exercise sessions. The examination usually lasts two hours.

During the assessment, the use of books, notes, personal sheets of paper, or electronic devices of any kind is not permitted. The use of a basic calculator is allowed.

Registration for the examination and communication of the result by the teacher take place through AlmaEsami.

Students who have passed the written examination may, if they wish, request to also take an oral examination.

Regarding the assessment of learning, the use of AI is prohibited. Any use of AI constitutes a violation of academic integrity.

RECOMMENDATIONS FOR TAKING THE FINAL EXAMINATION: In accordance with the University Code of Ethics, students are reminded to act with the utmost integrity. Any activity aimed at improperly altering the outcome of the examination is prohibited, including, for example, cheating, plagiarism, accessing online teaching materials, or using unauthorized AI tools. In particular, merely possessing unauthorized equipment or materials during the examination will result in the immediate invalidation of the exam paper and the matter being reported to the relevant University offices. Any conduct in breach of this prohibition may lead to disciplinary proceedings or, where the conduct constitutes a criminal offence, to reports being made to the competent authorities.

Students with specific learning disorders, or temporary or permanent disabilities, are advised to contact the relevant University office in good time: https://site.unibo.it/studenti-con-disabilita-e-dsa/it . The office will propose any necessary accommodations to the students concerned; these accommodations must in any case be submitted to the teacher for approval at least 15 days in advance. The teacher will assess their appropriateness, also in relation to the learning outcomes of the course.

 

Teaching tools

Blackboard, slides, educational materials on Virtuale.     

Office hours

See the website of Roberta Nibbi