THREE-DIMENSIONAL MODELING

Course objectives

Learning goals General goals. The primary educational goal of teaching is to learn the main concepts and methods of probability. Students must also be able to apply the above methods and be able to interpret the results that derive from their solution. Knowledge and understanding. After attending the course the students know and understand the main probabilistic concepts (events, random variables, probability distributions) Applying knowledge and understanding. At the end of the course the students are able to formalize real problems and to solve them by applying the specific methods of the discipline. Making judgements. Students develop critical skills through the application of probabilistic theory. Communication skills. Students, through the study and performance of practical exercises, acquire the technical-scientific language of the discipline, which must be used appropriately in both the intermediate and final written tests and in the oral tests. Learning skills. Students who pass the exam have learned a method of analysis that allows them to tackle the subsequent teachings of statistical-probabilistic area.

Channel 1
LUISA BEGHIN Lecturers' profile

Program - Frequency - Exams

Course program
Course introduction. Space of the results. Algebra of the events. Limits of event sequences. Algebra definitions and properties. Sigma algebra. Combinatorc (permutations, combinations, simple and repeated dispositions). Different approaches to probability (classic, frequency, subjective). Probability axioms. First probability properties and theorems. Probability and continuity. Finite and complete additivity. 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. Definition of random variables. Discrete random variables. Uniform, Bernoulli, binomial, degenerate distributions. Geometric random variable, Poisson, hypergeometric distribution. Poisson distribution as a Binomial limit. Distribution function and its properties. Continuous random variables. Distributions: uniform, exponential, gamma, normal. Transformation of random variables (three methods). Expected values and their application to Bermoulli’s binomial and exponential random variables. Variance and moments.. Moments of transformations. Cebicev's inequality. Variance of standard and non-standard Normal r.v.'s. Exercises on the transformations of random variables. 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. 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 calculus and applied mathematics
Books
ORSINGHER E., BEGHIN L., Introduzione alla Probabilità (2009), CAROCCI ED. DALL’AGLIO G., Calcolo delle Probabilità, III Ed., (2003) ZANICHELLI
Teaching mode
Lectures
Frequency
The attendance of lectures and classes is strongly suggested
Exam mode
Written and oral exams. Possible mid-term test
Lesson mode
Lectures
  • Academic year2025/2026
  • CourseStatistics, Finance and Actuarial sciences
  • CurriculumCurriculum unico
  • Year2nd year
  • Semester1st semester
  • SSDMAT/06
  • CFU9