The course of "Mathematics and principles of statistics" aims at providing students with the basic tools of mathematical analysis and statistics in order to be able to study, analyze and discuss real situations and phenomena through the use of mathematical models and statistical tools. With specific reference to the Dublin Descriptors, the learning objectives are set out as follows:
Knowledge understanding : at the end of the course, students will acquire specific knowledge on the methodologies of mathematical and statistical analysis to read, describe, specify and interpret a real phenomenon through technical tools of mathematical and statistical nature. With reference to the topics of mathematical analysis, students will develop methodological knowledge and will be provided with the basic tools to study linear and transcendental functions both through the study of limits and of differential calculus; they will also be able to elaborate real problems through the use of linear algebra and matrix calculus. As far as the notions of statistics are concerned, the aim of the course is to provide students with the methodological knowledge and the ability to use methods and tools for: a) the descriptive analysis of data; b) the introduction to the study of phenomena under conditions of uncertainty, through the notions of probability theory and random variables; c) the study of relationships between variables both from a descriptive point of view and an introduction to modelling through linear functions.
Applying knowledge and understanding: at the end of the course, students will have acquired methodological knowledge and analytical skills and will be able to autonomously interpret analyses and empirical researches on the most relevant areas of intervention, also applied, relevant and related to the degree course. Students will be able to: i) evaluate the results of empirical analyses; consider the appropriateness of the mathematical and statistical methodologies used; identify any limitations of the analyses carried out and consider the use of alternative approaches;.
Making judgements: the course is aimed at encouraging a critical approach to the use of different approaches, methods and techniques for mathematical-statistical modelling and data analysis for the interpretation of phenomena applied in the fields of interest of the degree course. Students: i) will develop critical skills on the use of various methods in relation to the analysis objectives of the phenomenon under study; ii) will be able to evaluate the contribution of a specific mathematical and data analysis methodology to the study of real phenomena, including complex ones; iii) will develop the ability to coherently integrate the contribution provided by quantitative analysis methods with the student's interdisciplinary skills.
Communication skills: students will have developed specific skills to communicate unambiguously and clearly the analysis scheme adopted for the empirical study and to model, through mathematical analysis and statistics, real phenomena. The ability to communicate effectively will also be validated through the verification of logical-argumentative and synthesis skills.
Learning skills: the teaching methodologies used during the course and the use of learning verification methods focused on the study of real functions and analysis of problems based on the study of empirical distributions will contribute to strengthen the students' ability of autonomy of judgement and the development of self-learning skills.
Introduction and review of basic mathematical notions. Natural numbers, sum and product of natural numbers. Neutral element and inverses. Relative numbers and rational numbers. Irrational and real numbers. Consistency and density of the real numbers. Units of measurement and conversion factors. Intervals. Operations. Scientific notation. Approximations. Equations and inequalities. Sets, inclusion relation, set representation with Eulero-Venn diagrams. Intersection, union, difference. Complement operation and cartesian product. Properties of the set operations. Functions and Properties of function. Injective, surjective, bijective functions. Real Functions of a Real Variable and their Geometrical Representation: graph, domain, codomain and image. Symmetry. Increasing and decreasing functions. Dependent and independent variables. Functional notation. Range and domain. The graph of a function. Function transformations: shifting, stretching, reflecting, sum, product and composition. Inverse functions. One-to-one and invertible functions. Determining the graph and equation of the inverse. Algebraic functions. Linear functions, quadratic functions, polynomial, power and rational functions. Transcendental functions: exponential and logarithmic functions. Introduction to trigonometric functions. Function composition. The inverse function. Limits and Continuity. Definition of limits, properties and calculating the limits. Limits of functions, continuity and asymptotes. Derivatives The Tangent to a Curve and the Derivative of a Function. Definition and calculation of derivative: algebraic and transcendental functions. The First and Second Derivatives. Left and right derivatives, higher order derivatives. Necessary condition of differentiability (with proof). Non differentiable functions.. Global and local maximum and minimum. Extreme value theorem. Differentiability and monotonicity. Differentiability and local extremes. Concavity and inflection points (with proofs). Necessary and sufficient conditions for the existence of inflection points (with proofs). De l’Hospital Theorem. Integration. Definite integration. Integrable and non integrable functions. Properties of the definite integral. Indefinite integration and antiderivatives. The fundamental theorem of calculus. Integration formulas. Integration by part and substitution.
Linear algebra. Vectors, vector spaces, geometric representation of vectors, linear dependence and independence. Matrices and properties. Determinants. Properties of determinants. Rank of a matrix. Matrix operations. Systems of linear equations. Rouché-Capelli theorem. Cramer rule. Eigenvalues and eigenvectors. Statistics. Notions of descriptive statistics: variables, categories and statistical units. Frequency distribution and graphical representations. Position measures: mean, median and mode. Variability and shape of a distribution. Analysis of the association between two characters, the ordinary least squares (OLS) estimator. Introduction to the linear regression models. Introduction to probability. Notion of event. Probability distributions. Axioms of probability. Independent events and incompatible events. Conditional probability. Bayes theorem. Random variables and probability distributions. Introduction to statistical inference and the theory of hypothesis testing.
in English: - Lial, M. L., Hungerford, T. W., Holcomb, J. P., & Mullins, B. (2018). Mathematics with applications: in the management, natural, and social sciences. Pearson.
- Whitlock M.C., Schluter D (2020) Analysis of Biological Data (International Edition - Third Edition)
in Italian: - Cea D., Secondi L. (2022) Elementi di Statistica e Matematica per le scienze applicate. Libreria Universitaria editrice, in corso di stampa Bodine et al (2017) Matematica per le scienze della vita. UTET - Guerraggio A. (2018) Matematica per le scienze. Pearson - Monti, A. (2008). Introduzione alla statistica. -Slides of the course and exercises made available in the student's portal.
Introductory notions: reminders of numerical calculation. Measurement units and conversion factors. Operations. Scientific notation. Approximations. Equalities and inequalities. Percentages. Elements of analytical geometry: Cartesian coordinates. Lines and segments. Conical. Equations and inequalities. Concepts of set theory. Basic descriptive statistics and data analysis: distribution of a statistical character and its graphical representation. Measures of position, variability and shape of a distribution. Analysis of the association between two characters, the method of least squares and introduction to linear regression models. Notion of function and property. Algebraic functions: linear, quadratic, polynomial functions, power functions and rational functions. Transcendent functions: exponential and logarithmic functions. Introduction to trigonometric functions. The functional composition. functional inversion. The definition in parts. Introduction to modeling for discrete-time systems. Linear algebra: vectors, vector spaces, geometric representation of vectors, linear dependence and independence. Matrices and determinants. Rank of a matrix. Matrix operations. Systems of linear equations. Rouché-Capelli theorem. Cramer's rule. Eigenvalues and eigenvectors. Combinatorial calculus and elements of probability theory. Event concept. Probability distributions. Probability axioms. Independent and incompatible events. Conditional probability. Bayes theorem. Random variables and probability distributions. Introduction to statistical inference and hypothesis testing theory.
Definition of limit, properties and calculation of limits. Limits of functions, continuity and asymptotes. Qualitative study of functions. Definition and calculus of derivatives: algebraic functions and transcendental functions. Growth and decrease. Minimums and maximums. Concavity and convexity. Taylor development. Introduction to partial derivatives. Integral calculus: definition of integral, properties of the integral. Indefinite integral. Integration by parts, integration by substitution. Definite integrals.
- Cea D., Secondi L. (2022) Elementi di Statistica e Matematica per le scienze applicate. Libreria Universitaria editrice, in corso di stampa - Materiale didattico supplementare prodotto dal docente e fornito agli studenti a seguito di ogni lezione
Dates of beginning and end of teaching activities
At a distance
Objectives of the course
Università degli Studi della Tuscia - Rettorato, Via S.M. in Gradi n.4, 01100 Viterbo, ITALY - Tel. 0761.3571