Luigi RARITA' | MATHEMATICS AND STATISTICS

cod. 0212700177

MATHEMATICS AND STATISTICS

	0212700177
	DEPARTMENT OF MANAGEMENT & INNOVATION SYSTEMS
	EQF6
	BUSINESS MANAGEMENT
	2024/2025

CURRICULUM MANAGEMENT AND COMPUTER SCIENCE

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2023
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	SECS-S/01	10	60	LESSONS	BASIC COMPULSORY SUBJECTS

PROFESSORS

	LUIGI RARITA' T Curriculum
	MARIA PIA D'ARIENZO Curriculum

EXAMS

	Exam	Date	Session
	RARITA'	11/12/2024 - 15:00	SESSIONE ORDINARIA
	RARITA'	11/12/2024 - 15:00	SESSIONE DI RECUPERO

	Objectives
	THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF LINEAR ALGEBRA AND STATISTICS AND FURTHER ELEMENTS OF MATHEMATICAL ANALYSIS. THE STUDENTS WILL ACQUIRE: •THE RESULTS AND DEMONSTRATION TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT CALCULATION TOOLS; •THE FOLLOWING ELEMENTS OF MATHEMATICS AND STATISTICS: MATRICES AND LINEAR SYSTEMS, VECTOR SPACES, EIGENVALUES AND EIGENVECTORS, FUNCTIONS OF TWO REAL VARIABLES, DESCRIPTIVE STATISTICS, PROBABILITY, RANDOM VARIABLES, SOFTWARE TOOLS. THE STUDENT WILL BE ABLE TO: •APPLY THE DEFINITIONS, THEOREMS AND THE STUDIED METHODS OF LINEAR ALGEBRA, MATHEMATICAL ANALYSIS AND STATISTICS FOR THE EFFICIENT MANAGEMENT AND RESOLUTION OF MATHEMATICAL PROBLEMS; •PERFORM OPERATIONS WITH MATRICES, SOLVE LINEAR SYSTEMS, COMPUTE BASES OF VECTOR SPACES AND SUBSPACES, DIAGONALIZE A MATRIX, DETERMINE MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLES, ANALYZE AND INTERPRET DATA, USE THE PROBABILITY AND RANDOM VARIABLES THEORY, USE STATISTICAL TOOLS IN APPLICATION CONTEXTS.

THE COURSE AIMS AT THE ACQUISITION OF THE BASIC ELEMENTS OF LINEAR ALGEBRA AND STATISTICS AND FURTHER ELEMENTS OF MATHEMATICAL ANALYSIS.
THE STUDENTS WILL ACQUIRE:
•THE RESULTS AND DEMONSTRATION TECHNIQUES, AS WELL AS THE ABILITY TO USE THE RELEVANT CALCULATION TOOLS;
•THE FOLLOWING ELEMENTS OF MATHEMATICS AND STATISTICS: MATRICES AND LINEAR SYSTEMS, VECTOR SPACES, EIGENVALUES AND EIGENVECTORS, FUNCTIONS OF TWO REAL VARIABLES, DESCRIPTIVE STATISTICS, PROBABILITY, RANDOM VARIABLES, SOFTWARE TOOLS.
THE STUDENT WILL BE ABLE TO:
•APPLY THE DEFINITIONS, THEOREMS AND THE STUDIED METHODS OF LINEAR ALGEBRA, MATHEMATICAL ANALYSIS AND STATISTICS FOR THE EFFICIENT MANAGEMENT AND RESOLUTION OF MATHEMATICAL PROBLEMS;
•PERFORM OPERATIONS WITH MATRICES, SOLVE LINEAR SYSTEMS, COMPUTE BASES OF VECTOR SPACES AND SUBSPACES, DIAGONALIZE A MATRIX, DETERMINE MAXIMA AND MINIMA OF FUNCTION OF TWO VARIABLES, ANALYZE AND INTERPRET DATA, USE THE PROBABILITY AND RANDOM VARIABLES THEORY, USE STATISTICAL TOOLS IN APPLICATION CONTEXTS.

	Prerequisites
	FOR THE SUCCESSFUL ACHIEVEMENT OF THE PREDEFINED OBJECTIVES AND, IN PARTICULAR, FOR AN ADEQUATE UNDERSTANDING OF THE CONTENTS PROVIDED BY THE TEACHING, KNOWLEDGE RELATED TO REAL FUNCTIONS OF A REAL VARIABLE AND DIFFERENTIAL CALCULUS ARE PARTICULARLY USEFUL AND, THEREFORE, REQUIRED TO THE STUDENT. MANDATORY PREPARATORY TEACHINGS MATHEMATICS.

	Contents
	MATRICES. (HOURS LECTURE/EXERCISE/LABORATORY 3/3/0) DEFINITIONS AND PROPERTIES. REDUCED ROW ECHELON FORM MATRICES. DETERMINANT AND LAPLACE THEOREM. RANK OF A MATRIX. INVERSE OF A MATRIX. LINEAR SYSTEMS. (HOURS LECTURE/EXERCISE/LABORATORY 3/3/0) SYSTEMS OF LINEAR EQUATIONS: DEFINITION, ASSOCIATED MATRICES, COMPATIBILITY CONDITIONS. RESOLUTION TECHNIQUES FOR LINEAR SYSTEMS. VECTOR SPACES. (HOURS LECTURE/EXERCISE/WORKSHOP 3/3/0) VECTOR SPACES STRUCTURE. VECTOR SUBSPACES. LINEAR DEPENDENCE AND INDEPENDENCE. SYSTEMS OF GENERATORS AND BASES. DIMENSION OF A VECTOR SPACE. STEINITZ LEMMA AND BASE THEOREM. CARTESIAN AND PARAMETRIC REPRESENTATION OF A VECTOR SUBSPACE. DIAGONALIZATION. (HOURS LECTURE/EXERCISE/WORKSHOP 3/3/0) EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND CHARACTERISTIC EQUATION. EIGENSPACES AND RELATED PROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZABLE MATRICES. FUNCTIONS OF SEVERAL VARIABLES. (HOURS LECTURE/EXERCISE/WORKSHOP 3/3/0) BASICS ON TOPOLOGY. LIMITS AND CONTINUITY. PARTIAL DERIVATIVES. GRADIENT. DIRECTIONAL DERIVATIVES. SCHWARZ THEOREM. DIFFERENTIAL AND TOTAL DIFFERENTIAL THEOREM. RELATIVE MAXIMA AND MINIMA FOR FUNCTIONS OF TWO VARIABLES. NECESSARY CONDITION OF THE FIRST ORDER AND SUFFICIENT CONDITION OF THE ECOND ORDER FOR RELATIVE MAXIMA AND MINIMA. STATISTICAL ANALYSIS OF DATA. (HOURS LECTURE/EXERCISE/LABORATORY 7/8/0). STATISTICAL SURVEYS. FREQUENCY DISTRIBUTIONS. POSITION AND DISPERSION INDICES. VARIABILITY INDICES. SHAPE OF A DISTRIBUTION. CORRELATION BETWEEN VARIABLES. LEAST SQUARES METHOD. REGRESSION LINE. CALCULATION OF PROBABILITIES. (HOURS LECTURE/EXERCISE/LABORATORY 4/3/0). RANDOM EXPERIMENTS, SAMPLE SPACE, EVENTS. BASICS ON COMBINATORIAL CALCULUS. PROBABILITY: CLASSICAL, STATISTICAL, SUBJECTIVE AND AXIOMATIC DEFINITIONS. UNION AND INTERSECTION EVENTS. LOGICAL SUM AND LOGICAL PRODUCT BETWEEN EVENTS. CONDITIONAL PROBABILITY. TOTAL PROBABILITY THEOREM. BAYES THEOREM. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. (HOURS LECTURE/EXERCISE/LABORATORY 4/4/0). RANDOM VARIABLES: GENERALITIES AND MAIN PROPERTIES; CONTINUOUS AND DISCRETE RANDOM VARIABLES. PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES: BINOMIAL AND POISSON DISTRIBUTION. PROBABILITY DENSITIES FOR CONTINUOUS RANDOM VARIABLES: UNIFORM AND EXPONENTIAL DISTRIBUTION; STANDARDIZED NORMAL DISTRIBUTION; GAUSSIAN DISTRIBUTION. TOTAL HOURS LECTURE/EXERCISE/LABORATORY 30/30/0

Contents

MATRICES.
(HOURS LECTURE/EXERCISE/LABORATORY 3/3/0)
DEFINITIONS AND PROPERTIES. REDUCED ROW ECHELON FORM MATRICES. DETERMINANT AND LAPLACE THEOREM. RANK OF A MATRIX. INVERSE OF A MATRIX.

LINEAR SYSTEMS.
(HOURS LECTURE/EXERCISE/LABORATORY 3/3/0)
SYSTEMS OF LINEAR EQUATIONS: DEFINITION, ASSOCIATED MATRICES, COMPATIBILITY CONDITIONS. RESOLUTION TECHNIQUES FOR LINEAR SYSTEMS.

VECTOR SPACES.
(HOURS LECTURE/EXERCISE/WORKSHOP 3/3/0)
VECTOR SPACES STRUCTURE. VECTOR SUBSPACES. LINEAR DEPENDENCE AND INDEPENDENCE. SYSTEMS OF GENERATORS AND BASES. DIMENSION OF A VECTOR SPACE. STEINITZ LEMMA AND BASE THEOREM. CARTESIAN AND PARAMETRIC REPRESENTATION OF A VECTOR SUBSPACE.

DIAGONALIZATION.
(HOURS LECTURE/EXERCISE/WORKSHOP 3/3/0)
EIGENVALUES AND EIGENVECTORS: DEFINITIONS, CHARACTERISTIC POLYNOMIAL AND CHARACTERISTIC EQUATION. EIGENSPACES AND RELATED PROPERTIES. ALGEBRAIC AND GEOMETRIC MULTIPLICITY. DIAGONALIZABLE MATRICES.

FUNCTIONS OF SEVERAL VARIABLES.
(HOURS LECTURE/EXERCISE/WORKSHOP 3/3/0)
BASICS ON TOPOLOGY. LIMITS AND CONTINUITY. PARTIAL DERIVATIVES. GRADIENT. DIRECTIONAL DERIVATIVES. SCHWARZ THEOREM. DIFFERENTIAL AND TOTAL DIFFERENTIAL THEOREM. RELATIVE MAXIMA AND MINIMA FOR FUNCTIONS OF TWO VARIABLES. NECESSARY CONDITION OF THE FIRST ORDER AND SUFFICIENT CONDITION OF THE ECOND ORDER FOR RELATIVE MAXIMA AND MINIMA.

STATISTICAL ANALYSIS OF DATA.
(HOURS LECTURE/EXERCISE/LABORATORY 7/8/0).
STATISTICAL SURVEYS. FREQUENCY DISTRIBUTIONS. POSITION AND DISPERSION INDICES. VARIABILITY INDICES. SHAPE OF A DISTRIBUTION. CORRELATION BETWEEN VARIABLES. LEAST SQUARES METHOD. REGRESSION LINE.

CALCULATION OF PROBABILITIES.
(HOURS LECTURE/EXERCISE/LABORATORY 4/3/0).
RANDOM EXPERIMENTS, SAMPLE SPACE, EVENTS. BASICS ON COMBINATORIAL CALCULUS. PROBABILITY: CLASSICAL, STATISTICAL, SUBJECTIVE AND AXIOMATIC DEFINITIONS. UNION AND INTERSECTION EVENTS. LOGICAL SUM AND LOGICAL PRODUCT BETWEEN EVENTS. CONDITIONAL PROBABILITY. TOTAL PROBABILITY THEOREM. BAYES THEOREM.

RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS.
(HOURS LECTURE/EXERCISE/LABORATORY 4/4/0).
RANDOM VARIABLES: GENERALITIES AND MAIN PROPERTIES; CONTINUOUS AND DISCRETE RANDOM VARIABLES. PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES: BINOMIAL AND POISSON DISTRIBUTION. PROBABILITY DENSITIES FOR CONTINUOUS RANDOM VARIABLES: UNIFORM AND EXPONENTIAL DISTRIBUTION; STANDARDIZED NORMAL DISTRIBUTION; GAUSSIAN DISTRIBUTION.

TOTAL HOURS LECTURE/EXERCISE/LABORATORY 30/30/0

	Teaching Methods
	THE TEACHING INCLUDES THEORETICAL LECTURES IN THE CLASSROOM FOR A TOTAL OF 30 HOURS AND CLASSROOM EXERCISES FOR A TOTAL OF 30 HOURS. ATTENDANCE OF CLASSROOM LECTURES AND EXERCISES, ALTHOUGH NOT MANDATORY, IS STRONGLY RECOMMENDED FOR THE FULL ACHIEVEMENT OF THE LEARNING OBJECTIVES.

	Verification of learning
	THE EXAMINATION IS DESIGNED TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE LECTURES; MASTERY OF THE MATHEMATICAL/STATISTICAL LANGUAGE IN THE WRITTEN AND ORAL EXAM; ABILITY TO PROVE THEOREMS AND ARGUE THE INTERPRETATION OF STATISTICAL DATA; ABILITY TO SOLVE EXERCISES; ABILITY TO IDENTIFY AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN SOLVING AN EXERCISE; AND ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO SOLVE DIFFERENT PROBLEMS WITH RESPECT TO THOSE PRESENTED DURING THE EXERCISES LECTURES. THE EXAMINATION, NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES, CONSISTS OF A WRITTEN TEST, PREPARATORY TO THE ORAL TEST, AND AN ORAL DISCUSSION. THE WRITTEN TEST INVOLVES THE RESOLUTION OF PROBLEMS SIMILAR TO THOSE PROPOSED DURING THE EXERCISES LECTURES. THE OUTCOME OF THE WRITTEN TEST IS “PASSED” OR “FAILED”. STUDENTS WHO PASS THE WRITTEN TEST WILL HAVE TO TAKE AND PASS THE ORAL TEST. THE ORAL TEST IS DESIGNED TO ASCERTAIN THE DEGREE OF KNOWLEDGE OF ALL TOPICS COVERED IN THE TEACHING, AND COVERS DEFINITIONS, STATEMENTS AND DEMONSTRATIONS OF THEOREMS, CRITICAL ANALYSIS OF STATISTICAL DATA AND PHENOMENA, AND RESOLUTION OF EXERCISES. THE FINAL GRADE, EXPRESSED IN THIRTIETHS WITH POSSIBLE LAUDE, IS DETERMINED AFTER THE OUTCOME OF THE ORAL DISCUSSION. IN EVALUATING THE EXAMINATION, ACCOUNT WILL BE TAKEN NOT ONLY OF KNOWLEDGE OF THE SUBJECT MATTER, BUT ALSO OF EXPOSITORY ABILITY, ACCURACY OF LANGUAGE, AND THE ABILITY TO USE CRITICALLY THE MATHEMATICAL/STATISTICAL ACQUIRED TOOLS. LAUDE WILL BE AWARDED TO STUDENTS WHO DEMONSTRATE EXCELLENT KNOWLEDGE OF COURSE CONTENTS, OPTIMAL EXPOSITORY SKILLS, AND HIGH MATURITY IN APPLYING THE KNOWLEDGE GAINED TO SOLVE PROBLEMS NOT ADDRESSED DURING CLASSROOM LECTURES.

Verification of learning

THE EXAMINATION IS DESIGNED TO ASSESS: KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED DURING THE LECTURES; MASTERY OF THE MATHEMATICAL/STATISTICAL LANGUAGE IN THE WRITTEN AND ORAL EXAM; ABILITY TO PROVE THEOREMS AND ARGUE THE INTERPRETATION OF STATISTICAL DATA; ABILITY TO SOLVE EXERCISES; ABILITY TO IDENTIFY AND APPLY THE MOST APPROPRIATE AND EFFICIENT METHODS IN SOLVING AN EXERCISE; AND ABILITY TO APPLY THE ACQUIRED KNOWLEDGE TO SOLVE DIFFERENT PROBLEMS WITH RESPECT TO THOSE PRESENTED DURING THE EXERCISES LECTURES.
THE EXAMINATION, NECESSARY TO ASSESS THE ACHIEVEMENT OF THE LEARNING OBJECTIVES, CONSISTS OF A WRITTEN TEST, PREPARATORY TO THE ORAL TEST, AND AN ORAL DISCUSSION.
THE WRITTEN TEST INVOLVES THE RESOLUTION OF PROBLEMS SIMILAR TO THOSE PROPOSED DURING THE EXERCISES LECTURES. THE OUTCOME OF THE WRITTEN TEST IS “PASSED” OR “FAILED”.
STUDENTS WHO PASS THE WRITTEN TEST WILL HAVE TO TAKE AND PASS THE ORAL TEST. THE ORAL TEST IS DESIGNED TO ASCERTAIN THE DEGREE OF KNOWLEDGE OF ALL TOPICS COVERED IN THE TEACHING, AND COVERS DEFINITIONS, STATEMENTS AND DEMONSTRATIONS OF THEOREMS, CRITICAL ANALYSIS OF STATISTICAL DATA AND PHENOMENA, AND RESOLUTION OF EXERCISES.
THE FINAL GRADE, EXPRESSED IN THIRTIETHS WITH POSSIBLE LAUDE, IS DETERMINED AFTER THE OUTCOME OF THE ORAL DISCUSSION. IN EVALUATING THE EXAMINATION, ACCOUNT WILL BE TAKEN NOT ONLY OF KNOWLEDGE OF THE SUBJECT MATTER, BUT ALSO OF EXPOSITORY ABILITY, ACCURACY OF LANGUAGE, AND THE ABILITY TO USE CRITICALLY THE MATHEMATICAL/STATISTICAL ACQUIRED TOOLS.
LAUDE WILL BE AWARDED TO STUDENTS WHO DEMONSTRATE EXCELLENT KNOWLEDGE OF COURSE CONTENTS, OPTIMAL EXPOSITORY SKILLS, AND HIGH MATURITY IN APPLYING THE KNOWLEDGE GAINED TO SOLVE PROBLEMS NOT ADDRESSED DURING CLASSROOM LECTURES.

	Texts
	C. D’APICE, T. DURANTE, R. MANZO: “VERSO L'ESAME DI MATEMATICA 2, RACCOLTA DI ESERCIZI CON SVOLGIMENTO”, MAGGIOLI EDITORE, APOGEO EDUCATION, 2015. S. BORRA, A. DI CIACCIO: “STATISTICA. METODOLOGIA PER LE SCIENZE ECONOMICHE E SOCIALI”, MCGRAW-HILL EDUCATION. G. CICCHITELLI, P. D’URSO, M. MINOZZO: “STATISTICA. PRINCIPI E METODI”, PEARSON. SUPPLEMENTARY TEACHING MATERIALS WILL BE AVAILABLE IN THE TEACHING SECTION WITHIN THE UNIVERSITY'S E-LEARNING AREA (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS IN THE COURSE VIA THE UNIQUE UNIVERSITY CREDENTIALS.

Texts

C. D’APICE, T. DURANTE, R. MANZO: “VERSO L'ESAME DI MATEMATICA 2, RACCOLTA DI ESERCIZI CON SVOLGIMENTO”, MAGGIOLI EDITORE, APOGEO EDUCATION, 2015.
S. BORRA, A. DI CIACCIO: “STATISTICA. METODOLOGIA PER LE SCIENZE ECONOMICHE E SOCIALI”, MCGRAW-HILL EDUCATION.
G. CICCHITELLI, P. D’URSO, M. MINOZZO: “STATISTICA. PRINCIPI E METODI”, PEARSON.
SUPPLEMENTARY TEACHING MATERIALS WILL BE AVAILABLE IN THE TEACHING SECTION WITHIN THE UNIVERSITY'S E-LEARNING AREA (HTTP://ELEARNING.UNISA.IT) ACCESSIBLE TO STUDENTS IN THE COURSE VIA THE UNIQUE UNIVERSITY CREDENTIALS.

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Luigi RARITA' | MATHEMATICS AND STATISTICS