MATHEMATICAL AND STATISTICAL METHODS FOR APPLICATIONS 1
Academic Year 2026/2027 - Teacher: VITTORIO ROMANOExpected Learning Outcomes
The aim of the course is to give the main tools for statistical investigations along with the study of advanced subjects for tackling problems of interest in mathematical physics, economics, industry and in general for applications arising from applied sciences. In particular, the course furnishes the background for mathematical analysis in economics and therefore suitable for student in the finance program.
Some aspects treated in the course are in any case relevant for those who want to teach Mathematics at the high schools.
In particular, the course aims to allow the student to acquire the following skills:
-knowledge and understanding: knowledge of results and fundamental methods in probability and statistics. Skill of understanding problems and to extract the major features. Skill of reading, understanding and analyzing a subject in the related literature and present it in a clear and accurate way.
-applying knowledge and understanding: skill of elaborating new example or solving novel theoretical exercise, looking for the most appropriate methods and applying them in an appropriate way.
-making judgements: to be able of devise proposals suited to correctly interprete complex problems in statistics and its applications. To be able to formulate autonomously adequate judgements on the applicability of numerical methods or statistical models to theoretical or real situations.
-communication skills: skills of presenting arguments, problems, ideas and solutions in mathematical terms with clarity and accuracy and with procedures suited for the audience, both in an oral and a written form. Skill of clearly motivating the choice of the strategy, method and contents, along with the employed computational tools.
-learning skills: reading and analyzing a subject in the literature involving applied mathematics. To tackle in an autonomuous way the systematic study of arguments not previously treated. To acquire a degree of autonomy such that the student can be able to start with an autonomuos reserach activity.
Course Structure
Mainly frontal lectures. Moreover, the theoretical acquired competencies will be applied in a laboratory where study cases will be tackled in a MATLAB environment.
If restrictions will be introduced because the COVID pandemic, le lectures will be given in a mixed way or only online and some changes could be introduced to assure the accomplishiments foreseen for the course.
Learning assessment may also be carried out online, should the conditions require it.
IMPORTANT: in order to guarantee equal opportunities to the students with handicap and/or any form of disability, such students may ask to talk to the teacher to program suitable actions. The interested students may also contact prof. Patrizia Daniele, the delegate in the Department of Mathematics and Computer Science for students with handicap and/or any form of disability.
Required Prerequisites
Attendance of Lessons
Detailed Course Content
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Reminds and complements of probability. Estimators. Characteristic functions. Various forms of large number law. Central limit theorem. Descriptive statistics. Inferential statistics. Hypothesis tests. Non parametrical estimators. Maximum likelihood method and normal correlation. Bayesian statistical inference. L
inear multiple regression. Non linear regression and applications to neuronal networks. Elements of programming in MatLab.
Textbook Information
V. Romano, Metodi Matematici per i Corsi di Ingegneria, CittàStudi
P. Baldi Calcolo delle probabilità e statistica, McGraw-Hill
R. Scozzafava Incertezza e probabilità, Zanichelli
D. C. Montgomery, G. C. Runger Applied statistics and probability for engineers, J. Wiley
L. Leuzzi, E. Marinari, G. Parisi, Calcolo delle probabilità. Un trattatelo per principianti volenterosi, Zanichelli
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Complements of probability. Characteristic functions. Large number law in several forms. Descriptive Statistics. Statistical inference. Tests of hypothesis. Maximum likelihood method. Normal correlation. Bayesian statistical inference. Simple and multiple linear regression. Nonlinear regression and applications to neuronal networks. | Notes of the lecturer |
| 2 | DISCRETE RANDOM VARIABLES: The binomial distribution. The hypergeometric distribution. The geometric distribution. The Poisson distribution. The negative binomial distribution. Multivariate random variables. Expectation and variance of a discrete random variable. The linear correlation coefficient. | Notes of the lecturer |
| 3 | CONTINUOUS RANDOM VARIABLES: Derivation of probability distributions. Expectation and variance of continuous random variables.Normal distributions. Gamma distributions. Exponential distributions. Chi-square distributions. Beta distributions. Weibull distributions.Introduction to reliability theory. Multivariate normal distributions. Review of measure theory. Characteristic functions. | Notes of the lecturer |
| 4 | LAW OF LARGE NUMBERS AND NORMAL APPRPXIMATION: Convergence in probability and the Weak Law of Large Numbers. Convergence indistribution. The Central Limit Theorem and normal approximation. The Strong Law of Large Numbers and the Ergodic Law of LargeNumbers. | Notes of the lecturer |
| 5 | DESCRIPTIVE STATISTICS: Data grouped by individual values (ungrouped data). Data grouped into classes (grouped data). | Notes of the lecturer |
| 6 | INFERENTIAL STATISTICS:Point estimation. Student’s t-distribution and Cochran's theorem. Confidence interval estimation. Confidence intervals for the mean,variance, and proportion. | Notes of the lecturer |
| 7 | HYPOTHESIS TESTING:General principles of hypothesis testing. Tests for the mean: known-variance and unknown-variance cases. Tests for the variance. Tests for a proportion. Tests for the mean in paired populations: known-variance and unknown-variance cases. Tests for the variance in pairedpopulations. Tests for proportions in paired populations. Chi-square test. Goodness-of-fit testing for a theoretical probability distribution.Chi-square test of independence. | Notes of the lecturer |
| 8 | Linear Regression and Analysis of Variance (ANOVA): Simple linear regression. Properties of residuals and assessment of the goodness-of-fit of linear regression models. Multiple linear regression. Regression from a machine learning perspective. Introduction toartificial neural networks. One-factor experimental designs. Two-factor experimental designs. | Notes of the lecturer |
| 9 | THEORY OF ESTIMATORS:Method of moments estimators. Maximum likelihood estimation. Bivariate normal correlation. Introduction to Bayesian statisticalinference. | Notes of the lecturer |
| 10 | MATLAB programming | Notes of the lecturer |
Learning Assessment
Learning Assessment Procedures
If necessary the exam will be online.