Scienze Biologiche | MATHEMATICS AND STATISTICS

cod. 0512800024

MATHEMATICS AND STATISTICS

	0512800024
	DEPARTMENT OF CHEMISTRY AND BIOLOGY "ADOLFO ZAMBELLI"
	EQF6
	BIOLOGICAL SCIENCES
	2022/2023

PERCORSO SHARED STUDY PLAN

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2016
	FULL ACADEMIC YEAR

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
1	MATEMATICA E STATISTICA (MODULO A)
	MAT/06	5	40	LESSONS	BASIC COMPULSORY SUBJECTS
2	MATEMATICA E STATISTICA (MODULO B)
	MAT/05	5	40	LESSONS	BASIC COMPULSORY SUBJECTS

PROFESSORS

	BARBARA MARTINUCCI1 T Curriculum
	ALESSANDRA MEOLI2 Curriculum

EXAMS

Exam	Date	Session
MATEMATICA E STATISTICA	27/06/2023 - 09:00	SESSIONE ORDINARIA
MATEMATICA E STATISTICA MOD A	27/06/2023 - 09:00	SESSIONE ORDINARIA
MOD. B - SOLO SE GIÀ SUPERATO MODULO A	27/06/2023 - 09:00	SESSIONE ORDINARIA

	Objectives
	THE AIM OF THE COURSE IS TO PROVIDE THE BASIC THEORY AND METHODS OF MATHEMATICAL ANALYSIS, PROBABILITY AND STATISTICS, NECESSARY FOR (I) THE ANALYTICAL UNDERSTANDING OF REAL WORLD PHENOMENA, (II) THE REPRESENTATION AND MODELING OF NATURAL PHENOMENA, (III) THE MANAGEMENT, ANALYSIS AND INTERPRETATION OF EXPERIMENTAL DATA RELATED TO BIOLOGICAL PHENOMENA. THE COURSE AIMS TO MAKE STUDENTS AWARE OF THE FORMALISM AND SPECIFIC TERMINOLOGY OF MATHEMATICS, SO THAT THEY WILL BE ABLE TO FACE PROBLEMS WITH A RIGOROUS AND LOGICALLY COHERENT APPROACH AND TO CORRECTLY ARGUE THE PROPOSED SOLUTIONS. KNOWLEDGE AND UNDERSTANDING THE COURSE AIMS TO LET THE STUDENT ACQUIRE THE FUNDAMENTAL NOTIONS OF MATHEMATICAL ANALYSIS, PROBABILITY AND STATISTICS. IN PARTICULAR, THE STUDENT WILL ACQUIRE KNOWLEDGE ON: - THE BASIC NOTIONS OF MATHEMATICAL ANALYSIS IN TERMS OF LIMITS, DERIVATIVES AND INTEGRALS OF REAL FUNCTIONS; - THE BASIC CONCEPTS OF PROBABILITY AND THE MAIN RANDOM VARIABLES; - THE BASIC STATISTICAL TOOLS AND ARGUMENTS OF HYPOTHESIS TESTING. APPLYING KNOWLEDGE AND UNDERSTANDING THE STUDENT WILL BE ABLE TO (I) APPLY THE KNOWLEDGE ACQUIRED TO ANALYZE, UNDERSTAND AND SOLVE PROBLEMS, (II) MODEL RANDOM PHENOMENA, (III) BUILD MATHEMATICAL MODELS OF A NATURAL PHENOMENON (IV) SYNTHESIZE AND INTERPRET EXPERIMENTAL DATA RELATED TO BIOLOGICAL PHENOMENA. TRANSVERSAL SKILLS THE STUDENT WILL BE ABLE (I) TO ADDRESS PROBLEMS BY IDENTIFYING THE HYPOTHESIS, IDENTIFYING THE MOST EFFECTIVE RESOLUTIVE PATH AND FORMALIZING THIS THROUGH ALGEBRIC AND GRAPHIC MODELS (PROBLEM SOLVING), (II) TO ACQUIRE AND COMMUNICATE THE INFORMATION OBTAINED FROM THE ANALYSIS OF A PROBLEM (EFFECTIVE COMMUNICATION), (III) TO ASSESS THE COHERENCE OF THE LOGICAL REASONING USED IN A PROOF (AUTONOMY OF JUDGMENT).

THE AIM OF THE COURSE IS TO PROVIDE THE BASIC THEORY AND METHODS OF MATHEMATICAL ANALYSIS, PROBABILITY AND STATISTICS, NECESSARY FOR (I) THE ANALYTICAL UNDERSTANDING OF REAL WORLD PHENOMENA, (II) THE REPRESENTATION AND MODELING OF NATURAL PHENOMENA, (III) THE MANAGEMENT, ANALYSIS AND INTERPRETATION OF EXPERIMENTAL DATA RELATED TO BIOLOGICAL PHENOMENA.

THE COURSE AIMS TO MAKE STUDENTS AWARE OF THE FORMALISM AND SPECIFIC TERMINOLOGY OF MATHEMATICS, SO THAT THEY WILL BE ABLE TO FACE PROBLEMS WITH A RIGOROUS AND LOGICALLY COHERENT APPROACH AND TO CORRECTLY ARGUE THE PROPOSED SOLUTIONS.

KNOWLEDGE AND UNDERSTANDING

THE COURSE AIMS TO LET THE STUDENT ACQUIRE THE FUNDAMENTAL NOTIONS OF MATHEMATICAL ANALYSIS, PROBABILITY AND STATISTICS. IN PARTICULAR, THE STUDENT WILL ACQUIRE KNOWLEDGE ON:
- THE BASIC NOTIONS OF MATHEMATICAL ANALYSIS IN TERMS OF LIMITS, DERIVATIVES AND INTEGRALS OF REAL FUNCTIONS;
- THE BASIC CONCEPTS OF PROBABILITY AND THE MAIN RANDOM VARIABLES;
- THE BASIC STATISTICAL TOOLS AND ARGUMENTS OF HYPOTHESIS TESTING.

APPLYING KNOWLEDGE AND UNDERSTANDING

THE STUDENT WILL BE ABLE TO (I) APPLY THE KNOWLEDGE ACQUIRED TO ANALYZE, UNDERSTAND AND SOLVE PROBLEMS, (II) MODEL RANDOM PHENOMENA, (III) BUILD MATHEMATICAL MODELS OF A NATURAL PHENOMENON (IV) SYNTHESIZE AND INTERPRET EXPERIMENTAL DATA RELATED TO BIOLOGICAL PHENOMENA.

TRANSVERSAL SKILLS

THE STUDENT WILL BE ABLE (I) TO ADDRESS PROBLEMS BY IDENTIFYING THE HYPOTHESIS, IDENTIFYING THE MOST EFFECTIVE RESOLUTIVE PATH AND FORMALIZING THIS THROUGH ALGEBRIC AND GRAPHIC MODELS (PROBLEM SOLVING), (II) TO ACQUIRE AND COMMUNICATE THE INFORMATION OBTAINED FROM THE ANALYSIS OF A PROBLEM (EFFECTIVE COMMUNICATION), (III) TO ASSESS THE COHERENCE OF THE LOGICAL REASONING USED IN A PROOF (AUTONOMY OF JUDGMENT).

	Prerequisites
	THE STUDENT MUST HAVE A SOLID KNOWLEDGE OF THE NOTIONS OF ALGEBRA AND ANALYTICAL GEOMETRY LEARNED IN THE SECONDARY SCHOOL. IN PARTICULAR, IT IS REQUIRED THE KNOWLEDGE OF FIRST AND SECOND DEGREE EQUATIONS AND INEQUALITIES, FRACTIONAL AND IRRATIONAL INEQUALITIES, SYSTEMS OF EQUATIONS, PROPERTIES OF POWERS AND LOGARITHMS.

	Contents
	MODULE A (5 CFU=40 H) ELEMENTS OF ARITHMETIC NUMERIC SETS. REAL NUMBERS. SCIENTIFIC NOTATION: MANTISSA AND EXPONENT, OPERATIONS IN SCIENTIFIC NOTATION. APPROXIMATIONS, ESTIMATED VALUE AND ABSOLUTE ERROR. ERROR PROPAGATION. PERCENTAGES. ELEMENTS OF MATHEMATICAL ANALYSIS REAL FUNCTIONS OF A REAL VARIABLE: BOUNDED, SYMMETRIC, MONOTONE, PERIODIC FUNCTIONS. GRAPHS OF FUNCTIONS. COMPOSITE FUNCTIONS, INVERTIBLE FUNCTIONS, INVERSE FUNCTIONS. ELEMENTARY FUNCTIONS, POWER FUNCTIONS, EXPONENTIAL AND LOGARITHMIC FUNCTIONS, TRIGONOMETRIC FUNCTIONS (SINE, COSINE, TANGENT). CONTINUOUS FUNCTIONS. LIMITS OF FUNCTIONS: DEFINITIONS AND FIRST PROPERTIES. UNIQUENESS OF THE LIMIT. LIMITS OF ELEMENTARY FUNCTIONS. OPERATIONS WITH LIMITS. INDETERMINATE FORMS. COMPARISON THEOREMS. HIERARCHY OF INFINITES AND INFINITESIMALS. ELEMENTS OF PROBABILITY SAMPLE SPACE, EVENTS. EVENT CLASS. COMBINATORICS. PERMUTATIONS, ARRANGEMENTS AND COMBINATIONS. CLASSICAL DEFINITION OF PROBABILITY. PROPERTIES OF PROBABILITY. CONDITIONAL PROBABILITY, LAW OF ALTERNATIVES, BAYES' THEOREM. INDEPENDENCE OF EVENTS. THE PROBLEM OF MEDICAL DIAGNOSIS. ELEMENTS OF STATISTICS DESCRIPTIVE STATISTICS. ABSOLUTE FREQUENCY AND CUMULATIVE ABSOLUTE FREQUENCY; RELATIVE FREQUENCY AND CUMULATIVE RELATIVE FREQUENCY; BAR CHART, HISTOGRAM AND PIE CHART. POSITION INDICES: SAMPLE MEAN, SAMPLE MODE AND SAMPLE MEDIAN. DISPERSION INDICES: SAMPLE VARIANCE AND COEFFICIENT OF VARIATION. SHAPE INDICES: ASYMMETRY AND KURTOSIS. BIVARIATE DATA. MODULE B (5 CFU=40 H) ELEMENTS OF GEOMETRY AND LINEAR ALGEBRA VECTORS IN THE PLANE AND IN SPACE: OPERATIONS ON VECTORS; LINEARLY INDEPENDENT VECTORS; SCALAR PRODUCT. EQUATION OF THE LINE, DIRECTIONAL VECTOR AND NORMAL VECTOR, ANGULAR COEFFICIENT. CARTESIAN AND PARAMETRIC EQUATION OF THE LINE SEGMENT. EQUATION OF THE PLANE IN SPACE. ELEMENTS OF MATHEMATICAL ANALYSIS CONTINUOUS FUNCTIONS: WEIERSTRASS THEOREM; THE INTERMEDIATE ZERO THEOREM. DIFFERENTIAL CALCULUS, DEFINITION OF DERIVATIVE, GEOMETRIC INTERPRETATION, EQUATION OF A TANGENT LINE TO THE GRAPH OF A FUNCTION. DERIVATIVES OF ELEMENTARY FUNCTIONS. RULES OF DERIVATION. DERIVATIVE OF A COMPOSITE FUNCTION. STATIONARY POINTS, LOCAL MAXIMA AND MINIMA, INTERVALS OF MONOTONICITY. SECOND DERIVATIVE, GEOMETRIC INTERPRETATION, CONVEXITY. STUDY OF THE GRAPH OF A FUNCTION. INTEGRAL CALCULUS: DEFINITE INTEGRAL. FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. INDEFINITE INTEGRAL. INTEGRALS OF ELEMENTARY FUNCTIONS. INTEGRATION METHODS: INTEGRATION BY PARTS AND BY SUBSTITUTION. ELEMENTS OF PROBABILITY RANDOM VARIABLES. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. DISTRIBUTION FUNCTION. DISCRETE PROBABILITY DISTRIBUTIONS: BERNOULLI AND BINOMIAL DISTRIBUTION. EXPECTED VALUE AND VARIANCE. ABSOLUTELY CONTINUOUS VARIABLES. PROBABILITY DENSITY FUNCTION. NORMAL DISTRIBUTION. CENTRAL LIMIT THEOREM. ELEMENTS OF STATISTICS LINEAR REGRESSION AND LEAST SQUARES METHOD. HYPOTHESIS TESTING. TYPES OF ERRORS AND LEVEL OF SIGNIFICANCE. HYPOTHESIS TEST ON THE MEAN WITH KNOWN VARIANCE. HYPOTHESIS TEST ON THE DIFFERENCE BETWEEN TWO MEANS IN THE CASE OF KNOWN VARIANCE. HYPOTHESIS TEST ON VARIANCE.

Contents

MODULE A (5 CFU=40 H)

ELEMENTS OF ARITHMETIC
NUMERIC SETS. REAL NUMBERS. SCIENTIFIC NOTATION: MANTISSA AND EXPONENT, OPERATIONS IN SCIENTIFIC NOTATION. APPROXIMATIONS, ESTIMATED VALUE AND ABSOLUTE ERROR. ERROR PROPAGATION. PERCENTAGES.

ELEMENTS OF MATHEMATICAL ANALYSIS
REAL FUNCTIONS OF A REAL VARIABLE: BOUNDED, SYMMETRIC, MONOTONE, PERIODIC FUNCTIONS. GRAPHS OF FUNCTIONS. COMPOSITE FUNCTIONS, INVERTIBLE FUNCTIONS, INVERSE FUNCTIONS. ELEMENTARY FUNCTIONS, POWER FUNCTIONS, EXPONENTIAL AND LOGARITHMIC FUNCTIONS, TRIGONOMETRIC FUNCTIONS (SINE, COSINE, TANGENT). CONTINUOUS FUNCTIONS. LIMITS OF FUNCTIONS: DEFINITIONS AND FIRST PROPERTIES. UNIQUENESS OF THE LIMIT. LIMITS OF ELEMENTARY FUNCTIONS. OPERATIONS WITH LIMITS. INDETERMINATE FORMS. COMPARISON THEOREMS. HIERARCHY OF INFINITES AND INFINITESIMALS.

ELEMENTS OF PROBABILITY
SAMPLE SPACE, EVENTS. EVENT CLASS. COMBINATORICS. PERMUTATIONS, ARRANGEMENTS AND COMBINATIONS. CLASSICAL DEFINITION OF PROBABILITY. PROPERTIES OF PROBABILITY. CONDITIONAL PROBABILITY, LAW OF ALTERNATIVES, BAYES' THEOREM. INDEPENDENCE OF EVENTS. THE PROBLEM OF MEDICAL DIAGNOSIS.

ELEMENTS OF STATISTICS
DESCRIPTIVE STATISTICS. ABSOLUTE FREQUENCY AND CUMULATIVE ABSOLUTE FREQUENCY; RELATIVE FREQUENCY AND CUMULATIVE RELATIVE FREQUENCY; BAR CHART, HISTOGRAM AND PIE CHART. POSITION INDICES: SAMPLE MEAN, SAMPLE MODE AND SAMPLE MEDIAN. DISPERSION INDICES: SAMPLE VARIANCE AND COEFFICIENT OF VARIATION. SHAPE INDICES: ASYMMETRY AND KURTOSIS. BIVARIATE DATA.

MODULE B (5 CFU=40 H)

ELEMENTS OF GEOMETRY AND LINEAR ALGEBRA
VECTORS IN THE PLANE AND IN SPACE: OPERATIONS ON VECTORS; LINEARLY INDEPENDENT VECTORS; SCALAR PRODUCT. EQUATION OF THE LINE, DIRECTIONAL VECTOR AND NORMAL VECTOR, ANGULAR COEFFICIENT. CARTESIAN AND PARAMETRIC EQUATION OF THE LINE SEGMENT. EQUATION OF THE PLANE IN SPACE.

ELEMENTS OF MATHEMATICAL ANALYSIS
CONTINUOUS FUNCTIONS: WEIERSTRASS THEOREM; THE INTERMEDIATE ZERO THEOREM. DIFFERENTIAL CALCULUS, DEFINITION OF DERIVATIVE, GEOMETRIC INTERPRETATION, EQUATION OF A TANGENT LINE TO THE GRAPH OF A FUNCTION. DERIVATIVES OF ELEMENTARY FUNCTIONS. RULES OF DERIVATION. DERIVATIVE OF A COMPOSITE FUNCTION. STATIONARY POINTS, LOCAL MAXIMA AND MINIMA, INTERVALS OF MONOTONICITY. SECOND DERIVATIVE, GEOMETRIC INTERPRETATION, CONVEXITY. STUDY OF THE GRAPH OF A FUNCTION. INTEGRAL CALCULUS: DEFINITE INTEGRAL. FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS. INDEFINITE INTEGRAL. INTEGRALS OF ELEMENTARY FUNCTIONS. INTEGRATION METHODS: INTEGRATION BY PARTS AND BY SUBSTITUTION.

ELEMENTS OF PROBABILITY
RANDOM VARIABLES. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. DISTRIBUTION FUNCTION. DISCRETE PROBABILITY DISTRIBUTIONS: BERNOULLI AND BINOMIAL DISTRIBUTION. EXPECTED VALUE AND VARIANCE. ABSOLUTELY CONTINUOUS VARIABLES. PROBABILITY DENSITY FUNCTION. NORMAL DISTRIBUTION. CENTRAL LIMIT THEOREM.

ELEMENTS OF STATISTICS
LINEAR REGRESSION AND LEAST SQUARES METHOD. HYPOTHESIS TESTING. TYPES OF ERRORS AND LEVEL OF SIGNIFICANCE. HYPOTHESIS TEST ON THE MEAN WITH KNOWN VARIANCE. HYPOTHESIS TEST ON THE DIFFERENCE BETWEEN TWO MEANS IN THE CASE OF KNOWN VARIANCE.
HYPOTHESIS TEST ON VARIANCE.

	Teaching Methods
	THE COURSE REQUIRES 80 HOURS OF TEACHING IN THE CLASSROOM DURING WHICH THEORETICAL TOPICS AND EXERCISES WILL BE DISCUSSED. THE EXERCISES REPRESENT AN INTEGRAL PART OF THE SCHEDULED LESSON.

	Verification of learning
	THE EXAM CONSISTS OF TWO TESTS, ONE FOR EACH OF THE TWO MODULES. TO VERBALIZE THE FINAL MARK IT IS NECESSARY TO PASS THE EXAMS RELATED TO THE TWO MODULES. THE EXAMS OF THE TWO MODULES MUST BE PASSED WITHIN THE SAME ACADEMIC YEAR. THE RESULTING VOTE IS GIVEN BY THE AVERAGE OF THE VOTES OBTAINED IN THE TWO FORMS. TO PASS EACH MODULE EXAM IT IS NECESSARY TO KNOW THE RESULTS OF THE THEORY PRESENTED IN THE LESSONS AND TO BE ABLE TO SOLVE THE EXERCISES. THE EXAMINATION OF EACH MODULE CONSISTS OF A WRITTEN TEST LASTING 2 HOURS AND CONTAINING EXERCISES. THE TEST IS PASSED IF YOU OBTAIN AT LEAST 18 POINTS OUT OF A TOTAL OF 30.

Verification of learning

THE EXAM CONSISTS OF TWO TESTS, ONE FOR EACH OF THE TWO MODULES. TO VERBALIZE THE FINAL MARK IT IS NECESSARY TO PASS THE EXAMS RELATED TO THE TWO MODULES. THE EXAMS OF THE TWO MODULES MUST BE PASSED WITHIN THE SAME ACADEMIC YEAR. THE RESULTING VOTE IS GIVEN BY THE AVERAGE OF THE VOTES OBTAINED IN THE TWO FORMS.

TO PASS EACH MODULE EXAM IT IS NECESSARY TO KNOW THE RESULTS OF THE THEORY PRESENTED IN THE LESSONS AND TO BE ABLE TO SOLVE THE EXERCISES. THE EXAMINATION OF EACH MODULE CONSISTS OF A WRITTEN TEST LASTING 2 HOURS AND CONTAINING EXERCISES. THE TEST IS PASSED IF YOU OBTAIN AT LEAST 18 POINTS OUT OF A TOTAL OF 30.

	Texts
	SERGIO INVERNIZZI, MAURIZIO RINALDI, FEDERICO COMOGLIO “MODULI DI MATEMATICA E STATISTICA”, ZANICHELLI, 2018 MARCO ABATE “MATEMATICA E STATISTICA”, LE BASI PER LE SCIENZE DELLA VITA, MC GRAW HILL EDUCATION, 2013