Antonio DI CRESCENZO | PROBABILITY

cod. 0512300043

PROBABILITY

	0512300043
	DEPARTMENT OF MATHEMATICS
	EQF6
	MATHEMATICS
	2024/2025

PERCORSO SHARED STUDY PATHWAY

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/06	7	56	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

PROFESSORS

	ANTONIO DI CRESCENZO T Curriculum
	ALESSANDRA MEOLI Curriculum

EXAMS

Exam	Date	Session
PROBABILITA'	08/01/2025 - 14:00	SESSIONE ORDINARIA
PROBABILITA'	08/01/2025 - 14:00	SESSIONE DI RECUPERO
PROBABILITÀ	29/01/2025 - 09:00	SESSIONE ORDINARIA
PROBABILITÀ	29/01/2025 - 09:00	SESSIONE DI RECUPERO
PROBABILITÀ	19/02/2025 - 09:00	SESSIONE ORDINARIA
PROBABILITÀ	19/02/2025 - 09:00	SESSIONE DI RECUPERO

	Objectives
	LEARNING OUTCOMES: THE COURSE HAS THE PRIMARY AIM OF PROVIDING THE BASIC NOTIONS OF PROBABILITY THEORY. KNOWLEDGE AND UNDERSTANDING: IN-DEPTH KNOWLEDGE OF THE BASIC TOPICS OF PROBABILITY. ABILITY TO IDENTIFY A PROBABILISTIC MODEL AND UNDERSTAND ITS MAIN CHARACTERISTICS. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING: INDUCTIVE AND DEDUCTIVE REASONING ABILITY IN DEALING WITH PROBLEMS INVOLVING RANDOM PHENOMENA. ABILITY TO SCHEMATIZE A RANDOM PHENOMENON IN RIGOROUS TERMS, TO SET UP A PROBLEM AND SOLVE IT USING APPROPRIATE PROBABILITY TOOLS. MAKING JUDGEMENTS: THE STUDENT WILL BE ABLE TO FORMULATE AND SOLVE PROBLEMS THAT REQUIRE THE CONSTRUCTION OF A MATHEMATICAL MODEL BASED ON BASIC KNOWLEDGE OF PROBABILITY. FURTHERMORE, THE STUDENT WILL BE ABLE TO ADAPT THE FORMAL STRATEGIES SEEN DURING THE COURSE TO NEW CONTEXTS. COMMUNICATION SKILLS: WITH REFERENCE TO PHENOMENA SUBJECT TO UNCERTAINTY, THE STUDENT WILL BE ABLE TO: - FORMALLY DESCRIBE COMPLEX CONCEPTS REFERRING TO THESE PHENOMENA, - FULLY ILLUSTRATE THEIR FUNDAMENTAL PROPERTIES, - RIGOROUSLY ARGUE THE STRATEGIES FOR SOLVING RELATED PROBLEMS. LEARNING ABILITY: THE STUDENT WILL BE ABLE TO: - USE TRADITIONAL BIBLIOGRAPHIC TOOLS AND IT RESOURCES FOR INDEPENDENT STUDY; - UNDERSTAND AND INTERPRET COMPLEX TEXTS; - PROCEED WITH THE CONTINUOUS UPDATING OF ONE'S KNOWLEDGE, USING TECHNICAL AND SCIENTIFIC LITERATURE.

LEARNING OUTCOMES:
THE COURSE HAS THE PRIMARY AIM OF PROVIDING THE BASIC NOTIONS OF PROBABILITY THEORY.

KNOWLEDGE AND UNDERSTANDING:
IN-DEPTH KNOWLEDGE OF THE BASIC TOPICS OF PROBABILITY. ABILITY TO IDENTIFY A PROBABILISTIC MODEL AND UNDERSTAND ITS MAIN CHARACTERISTICS.

ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
INDUCTIVE AND DEDUCTIVE REASONING ABILITY IN DEALING WITH PROBLEMS INVOLVING RANDOM PHENOMENA. ABILITY TO SCHEMATIZE A RANDOM PHENOMENON IN RIGOROUS TERMS, TO SET UP A PROBLEM AND SOLVE IT USING APPROPRIATE PROBABILITY TOOLS.

MAKING JUDGEMENTS:
THE STUDENT WILL BE ABLE TO FORMULATE AND SOLVE PROBLEMS THAT REQUIRE THE CONSTRUCTION OF A MATHEMATICAL MODEL BASED ON BASIC KNOWLEDGE OF PROBABILITY. FURTHERMORE, THE STUDENT WILL BE ABLE TO ADAPT THE FORMAL STRATEGIES SEEN DURING THE COURSE TO NEW CONTEXTS.

COMMUNICATION SKILLS:
WITH REFERENCE TO PHENOMENA SUBJECT TO UNCERTAINTY, THE STUDENT WILL BE ABLE TO:
- FORMALLY DESCRIBE COMPLEX CONCEPTS REFERRING TO THESE PHENOMENA,
- FULLY ILLUSTRATE THEIR FUNDAMENTAL PROPERTIES,
- RIGOROUSLY ARGUE THE STRATEGIES FOR SOLVING RELATED PROBLEMS.

LEARNING ABILITY:
THE STUDENT WILL BE ABLE TO:
- USE TRADITIONAL BIBLIOGRAPHIC TOOLS AND IT RESOURCES FOR INDEPENDENT STUDY;
- UNDERSTAND AND INTERPRET COMPLEX TEXTS;
- PROCEED WITH THE CONTINUOUS UPDATING OF ONE'S KNOWLEDGE, USING TECHNICAL AND SCIENTIFIC LITERATURE.

	Prerequisites
	STUDENTS MUST HAVE ACQUIRED THE ABILITY TO DEVELOP LOGICAL-MATHEMATICAL REASONING, BASED ON NOTIONS OF COURSES OF THE FIRST TWO YEARS OF THE MATHEMATICS DEGREE. KNOWLEDGE OF MATHEMATICAL ANALYSIS I/II IS OFFICIALLY REQUIRED.

	Contents
	SAMPLE SPACE. PROBABILITY. PROBABILITY SPACE. CONDITIONAL PROBABILITY. INDEPENDENCE. RANDOM VARIABLES. DISTRIBUTION FUNCTION. MEAN, STANDARD DEVIATION, VARIANCE. DISCRETE, CONTINUOUS, AND SINGULAR RANDOM VARIABLES. RANDOM VECTORS. INDEPENDENCE. COVARIANCE AND CORRELATION. MOMENTS. MOMENT GENERATING FUNCTION. PROBABILITY GENERATING FUNCTION. CHEBYSHEV INEQUALITY. CONVERGENCE OF RANDOM VARIABLES. LAW OF LARGE NUMBERS. CENTRAL-LIMIT THEOREM.

Contents

SAMPLE SPACE. PROBABILITY. PROBABILITY SPACE. CONDITIONAL PROBABILITY. INDEPENDENCE. RANDOM VARIABLES. DISTRIBUTION FUNCTION. MEAN, STANDARD DEVIATION, VARIANCE. DISCRETE, CONTINUOUS, AND SINGULAR RANDOM VARIABLES. RANDOM VECTORS. INDEPENDENCE. COVARIANCE AND CORRELATION. MOMENTS. MOMENT GENERATING FUNCTION. PROBABILITY GENERATING FUNCTION. CHEBYSHEV INEQUALITY. CONVERGENCE OF RANDOM VARIABLES. LAW OF LARGE NUMBERS. CENTRAL-LIMIT THEOREM.

	Teaching Methods
	LECTURES AND CLASSROOM EXERCISES. INDEED, THE LESSONS WILL FOCUS ON THEORETICAL TOPICS CONSTANTLY SUPPORTED BY THE PRESENTATION OF EXERCISES.

	Verification of learning
	THE FINAL TEST IS AIMED AT EVALUATING AS A WHOLE THE KNOWLEDGE AND ABILITIES OF UNDERSTANDING THE CONCEPTS PRESENTED IN THE LESSONS, AS WELL AS THE ABILITY TO APPLY SUCH KNOWLEDGE IN THE FORMALIZATION OF PROBLEMS SUBJECT TO RANDOMNESS AND IN THEIR RESOLUTION THROUGH TOOLS OF PROBABILITY THEORY. THE EXAM CONSISTS OF A WRITTEN TEST (WITH A MARK OUT OF THIRTY), AIMED AT ASSESSING THE ABILITY TO SOLVE PROBLEMS, AND IN AN ORAL INTERVIEW (WITH A MARK OUT OF THIRTY) AIMED AT ASSESSING THE KNOWLEDGE ACQUIRED IN THE THEORETICAL ASPECTS OF THE DISCIPLINE. ACCESS TO THE ORAL TEST IS SUBJECT TO ACHIEVING SUFFICIENCY IN THE WRITTEN TEST. EACH EXERCISE ASSIGNED TO THE WRITTEN TEST IS WORTH 10 POINTS. THE FINAL GRADE IS DETERMINED BY THE AVERAGE OF THE GRADES OF THE TWO TESTS. PRAISE MAY BE AWARDED TO STUDENTS WHO SHOW THEY ARE ABLE TO APPLY THE ACQUIRED KNOWLEDGE AND SKILLS WITH ORIGINALITY, WHO THEREFORE ARE ABLE TO REASON INDEPENDENTLY, PROPOSING, FOR EXAMPLE, ALTERNATIVE PROOFS TO THOSE PRESENTED IN THE LESSONS.

Verification of learning

THE FINAL TEST IS AIMED AT EVALUATING AS A WHOLE THE KNOWLEDGE AND ABILITIES OF UNDERSTANDING THE CONCEPTS PRESENTED IN THE LESSONS, AS WELL AS THE ABILITY TO APPLY SUCH KNOWLEDGE IN THE FORMALIZATION OF PROBLEMS SUBJECT TO RANDOMNESS AND IN THEIR RESOLUTION THROUGH TOOLS OF PROBABILITY THEORY.

THE EXAM CONSISTS OF A WRITTEN TEST (WITH A MARK OUT OF THIRTY), AIMED AT ASSESSING THE ABILITY TO SOLVE PROBLEMS, AND IN AN ORAL INTERVIEW (WITH A MARK OUT OF THIRTY) AIMED AT ASSESSING THE KNOWLEDGE ACQUIRED IN THE THEORETICAL ASPECTS OF THE DISCIPLINE.
ACCESS TO THE ORAL TEST IS SUBJECT TO ACHIEVING SUFFICIENCY IN THE WRITTEN TEST. EACH EXERCISE ASSIGNED TO THE WRITTEN TEST IS WORTH 10 POINTS.
THE FINAL GRADE IS DETERMINED BY THE AVERAGE OF THE GRADES OF THE TWO TESTS.

PRAISE MAY BE AWARDED TO STUDENTS WHO SHOW THEY ARE ABLE TO APPLY THE ACQUIRED KNOWLEDGE AND SKILLS WITH ORIGINALITY, WHO THEREFORE ARE ABLE TO REASON INDEPENDENTLY, PROPOSING, FOR EXAMPLE, ALTERNATIVE PROOFS TO THOSE PRESENTED IN THE LESSONS.

	Texts
	- ROSS S.M. (2010) A FIRST COURSE IN PROBABILITY. PEARSON PRENTICE HALL. - ROSENTHAL J. (2006) A FIRST LOOK AT RIGOROUS PROBABILITY THEORY. WORLD SCIENTIFIC PUBLISHING. - CHUNG, K.L. (2012) ELEMENTARY PROBABILITY THEORY WITH STOCHASTIC PROCESSES. SPRINGER SCIENCE & BUSINESS MEDIA. - STROOCK, D.W. (2013) MATHEMATICS OF PROBABILITY. VOL. 149. AMERICAN MATHEMATICAL SOC.

	More Information
	THE ATTENDANCE IN TERM TIME IS RECOMMENDED. SOME EDUCATIONAL AIDS ARE AVAILABLE THROUGH THE TEAMS E-LEARNING PLATFORM EMAIL: ADICRESCENZO@UNISA.IT, AMEOLI@UNISA.IT

Lessons Timetable

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Antonio DI CRESCENZO | PROBABILITY