PROBABILITY

Course objectives

Learning goals The primary educational objective of the course is students' learning of the main theoretical aspects related to probability. Students must also be able to solve the analytical problems necessary to apply the aforementioned theoretical concepts. Knowledge and understanding. At the end of the course the students know and understand the main aspects related to the theory of probability and the main methods useful to solve the problems linked to the uncertainty. Applying knowledge and understanding. At the end of the course students are able to formalize problems related to uncertainty in terms of probabilistic problems and to apply the specific methods of the probability to solve them. They are also able to model real phenomena through remarkable probabilistic structures. Making judgements. Students develop critical skills through the application of theory to a wide range of probabilistic models. They also develop the critical sense through the comparison between alternative solutions to the same problem obtained using different methodological aspects. Communication skills. Students, through the study and the practical exercises, acquire the technical-scientific language of the probability, which must be properly used both in the intermediate and final written tests and in the oral tests. Learning skills. Students who pass the exam have learned the basic concepts of probability that allow them to deal with subsequent statistical area teaching (in particular the teaching of Statistical Inference).

Channel 1
COSTANTINO RICCIUTI Lecturers' profile

Program - Frequency - Exams

Course program
PART 1 (about 20 hours) Course introduction. Sample space. Algebra of the events. Limits of event sequences. Algebra definitions and properties. Sigma algebra. Combinatorics (permutations, combinations, simple and repeated dispositions). Different approaches to probability (classic, frequency, subjective). Probability axioms. First probability properties and theorems. Probability and continuity. Law of total probability. Boole's inequality. Conditional probability. Bayes' theorem. Independence between two events. Independence among n events. Theory and exercises on the various kinds of draws from an urn. PART 2(about 40 hours) Definition of random variables. Discrete random variables. Uniform, Bernoulli, binomial, degenerate, Poisson and geometric distributions. Poisson distribution as a Binomial limit. Distribution function and its properties. Continuous random variables. Uniform, exponential, gamma, normal distributions. Transformation of random variables (three methods). Expected values, variance and moments. Cebicev's inequality. Multidimensional random variables and their properties. Relations between random variables. Independence between two or n random variables. Conditional probability distribution (discrete and continuous case). Functions of multivariate random variables. Expected values of multidimensional random variables. Conditional expected values. Sum of independent random variables (Convolution). Examples: Sum of two gamma, exponential, Poisson and binomial r.v.'s. Distribution of the maximum and the minimum of n independent and dependent r.v.'s. PART 3 (about 10 hours) Moment generating functions and characteristic functions. Convergence of r.v.'s and random vectors: in distribution, in probability, almost surely. Central limit theorem and examples. The weak and strong laws of large numbers.
Prerequisites
Basic knowledge of mathematical analysis.
Books
LIBRI DI TESTO: 1) S. Ross, Calcolo delle probabilità 2) P. Baldi, Calcolo delle probabilità. 3) G. Dall'Aglio, Calcolo delle probabilità LETTURE CONSIGLIATE PER L'APPROFONDIMENTO: 1) L. Leuzzi, E. Marinari, G. Parisi, CALCOLO DELLE PROBABILITÀ 2) E. Orsingher, L. Beghin, Introduzione alla probabilità.
Teaching mode
Lectures and exercises, in blended mode.
Frequency
Attendance is optional, but strongly recommended.
Exam mode
Written exam consisting in solving exercises (in 2 hours and 30 minutes). Oral exam with theoretical questions. In general, the written test and the oral test have the same weight on the formulation of the final grade.
Bibliography
Books on Probability theory
Lesson mode
Lectures and exercises, in blended mode.
  • Lesson code1022318
  • Academic year2024/2025
  • CourseStatistics, Economics and Society
  • CurriculumSingle curriculum
  • Year2nd year
  • Semester1st semester
  • SSDMAT/06
  • CFU9
  • Subject areaStatistico - probabilistico