# Unit MATHEMATICS II AND STATISTICS

Course
Engineering management
Study-unit Code
A002895
Curriculum
In all curricula
CFU
12
Course Regulation
Coorte 2023
Offered
2023/24
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa integrata

### MATHEMATICS II

Code A002901 6 Luca Zampogni Luca Zampogni 54 ore - Luca Zampogni Base Matematica, informatica e statistica MAT/05 Obbligatorio (Required) English Functions of two real variables, gradient, directional derivatives, tangent plane and linear approximation, some differential calculus for functions of two real variables. First order differential equations. Linear systems of the first order differential equations. Exponential and logistic models for growth, some models in physics. 1)Calculus for Scientists and Engineersbrokate martin; manchanda pammy; siddiqi abul hasan, Springer ed2) Calculus for Business, Economics, Life Sciences, and Social SciencesBarnett, Raymond; Ziegler, Michael; Byleen, Karl; Stocker, Christopher, Pearson ed.3) Calculus: Early Transcendentals, J. Stewart, Cengage The main objective of the course is that of extending the basic concepts of Mathematical Analysis to the calculus in two real variables, to the effect of constructing and understanding more complex mathematical models in the applied sciences In order to be able to understand and reach the objectives of the course of Mathematical Analysis, the student must have gained all the knowledge and abilities related to topics of the first semester course program of Mathematics. The course is organized as follows: lectures on all subjects of the course, classroom exercises to prepare students for the written exams. Attendance: optional but strongly recommended The exam is made of both a written and oral test. The written test consists of the solution of four problems and has a duration of at most three hours. Its objectives are the following: - The understanding of the proposed problems; - The handling of mathematical instruments; - The interpretation of the results obtained. The oral test consists of a talk of about 20 minutes and is aimed at testing the degree of comprehension reached by the student and his skills in handling mathematical objects, with particular attention to his capacity of finding connections between the topics explained. Functions of two real variables: domain, range, graph. Level curves. Quadratic surfaces. Partial derivatives, gradient. Directional derivatives. Higher order partial derivatives. Tangent plane and linear approximation. Maxima and minima for functions of two real variables.Regular curves, line integralsMultiple integration. Fubini’s Theorem, change of variables, Jacobian.Vector fields. Irrotational and conservative vector fields. Line integrals of vector fields. Gauss-Green Theorem, Divergence Theorem, Stokes Theorem.Ordinary differential equation: terminology and notations, solution, order. Initial value problems. Existence and uniqueness of the solution. First-order equations.: The Malthusian growth model. First-order linear equations. The Bernoulli equation. The Verhulst model (the logistic growth model). The Gompertz model. Linear second order equations and applications.

### STATISTICS

Code A002903 6 Luca Scrucca Barbara Guardabascio (Codocenza)Luca Scrucca 27 ore (Codocenza) - Barbara Guardabascio27 ore - Luca Scrucca Base Matematica, informatica e statistica SECS-S/02 Obbligatorio (Required) ENGLISH In a world data driven it is more and more important let students acquire an operative ability in terms of data retrieval and analysis of data (strongly appreciated by the labour market) which allows them to evaluate and interpret more independently and effectively the numerous statistics daily provided by all types of mediaThe course of Statistics provides several insights to various facets of statistics and concepts required to solve various data science problems. It is an in-depth presentation of the main topics in statistical science with which any data scientist should be familiar including descriptive statistics, probability theory, statistical estimation and inference methods, linear and logistic regression models. Emphasis will be placed on intuitive and rigorous understanding of the fundamentals of statistics, accompanied by implementation of statistical methods to solve real world business problems. The practical part is focused on the computation and interpretation of empirical results, and it will be realized using the R software. Reading material on each course topic (handouts, slides, data sets, R scripts), will be made available to the students by the course instructors during the course. Suggested books are:Alan Agresti and Maria Kateri (2021), “Foundations of Statistics for Data Scientists”. CRC Press Taylor & Francis Group.Alan Agresti and Maria Kateri (2021), “R-Web-Appendix of Foundations of Statistics for Data Scientists”. Freely available at http://stat4ds.rwth-aachen.de/pdf/DS_R_webAppendix.pdf By the end of the course students should:Know the fundamental theoretical notions of statistical analysis and illustrate the basic techniques for organizing, summarizing and graphically representing a dataset. They will also be able to formulate and conduct simple inferential procedures and regression analyses on small datasets.Produce and interpret basic statistical analyses. The objectives of the course, thus, include the ability of recognizing the different types of data and selecting among several statistical tools the most appropriate for the problem at hand.Interpret, evaluate and compare linear regression models, summarize and communicate the results with suitable graphical representations. The statistical methods introduced in the course are strongly dependent on basic mathematical tools, therefore it is expected that students had successfully attended and completed the planned mathematics courses. The course is carried out through lectures and practical exercises using data. Techniques will be introduced by examples and described in mathematical formulas. Focus will be on the practical implementation of each technique, R coding and interpretation of results. Attending classes is strongly advised. Assessment will be through a written examination, containing both theoretical questions and numerical data analysis exercises. The examination is designed to test students’ analytical skills, interpretation and understanding of relevant issues presented during the course. Descriptive statisticsIntroduction to statistical data science. Type of data and variables. Summarizing univariate data: location and variability indices, graphical representations. Descriptive statistics for multivariate data: bivariate quantitative data, categorical data and contingency tables, graphical representations.Probability Definitions of probability, sample space and events, probability axioms. Marginal and conditional probability, independent events, theorem of total probability, Bayes theorem and its applications.Random variables, discrete and continuous probability distributions. Main probability distributions: Bernoulli, Binomial, Normal, Chi-Square, t-distribution, F-distribution.Sampling distributions and the central limit theorem.InferencePoint estimates and confidence intervals. Significance testing.Linear regression Introduction to statistical models. The simple linear regression model: ordinary least squares estimation of the regression coefficients and error variance. Inference on the regression coefficients of normal linear model. Model assessment. Predicting future observations. The multiple linear regression model.Logistic regressionThe logistic regression model for binary data. Maximum likelihood estimation and parameters interpretation: effects on probability and odds. Forecasting.
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