Studying at the University of Verona
Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.
Study Plan
This information is intended exclusively for students already enrolled in this course.If you are a new student interested in enrolling, you can find information about the course of study on the course page:
Laurea in Matematica applicata - Enrollment from 2025/2026The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.
1° Year
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2° Year activated in the A.Y. 2014/2015
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3° Year activated in the A.Y. 2015/2016
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Uno o due insegnamenti tra i seguenti per un totale di 12 cfu
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Modules | Credits | TAF | SSD |
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Modules | Credits | TAF | SSD |
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Uno o due insegnamenti tra i seguenti per un totale di 12 cfu
Modules | Credits | TAF | SSD |
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Legend | Type of training activity (TTA)
TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.
Probability (2014/2015)
Teaching code
4S02753
Credits
6
Language
Italian
Scientific Disciplinary Sector (SSD)
MAT/06 - PROBABILITY AND STATISTICS
The teaching is organized as follows:
Teoria
Esercitazioni
Learning outcomes
The course aims at introducing the Probability theory fundamentals in the more general framework of the Lebesgue measure theory. Particular attention is given to the analytical aspects of the axiomatic basis of Kolmogorov approach to modern Probability, e.g. the construction of general probability spaces, the definition of algebra, sigma-algebra, Borel sets, measurable functions, push-forward measures, etc.
The course is basically divided into two parts devoted to the definition and study of discrete, resp. continuous, random variables (r.v.).
The introduction to the fundamental concepts of the modern theory of probability is classical and based on the elements of combinatorics, the laws of set theory and on the propositional calculus fundamentals.
The approach to r.v. in the continuum is first developed in a strictly probabilistic framework, with references to some basic analytical aspects such as those of integral calculus (integration in R^n, Fubini's theorem, dominated convergence, etc.), the convolution of functions, Laplace and Fourier transforms, etc.
In a second step the probabilistic aspects are reviewed in the context of the theory of measure, especially concerning theorems of convergence for sequences of r.v., also including the central limit theorem.
During the entire course, lessons are always characterized by the presentation of examples and relevant problems. Additionally, the student are continously requested to solve exercises, of different difficulty, which are proposed by the teacher, as weel as by the tutors.
Program
Fundamentals of Probability with respect to the axiomatic approach à la Kolmogorov
Independent/incompatible events
Rudiments of combinatorics (eg, combinations, permutations)
Uniform probability spaces
Conditional probability
Experiments with repeated independent trials
Probabilistic definition of random variable (rv)
Discrete random variables with values in R^n
o distribution function
o density function (discrete)
o Joint laws (discrete), marginals and conditional independence
o Examples: Bernoulli, binomial, geometric, Poisson, etc.
o Mean, variance and covariance operators
o Index of correlation
o Moments of a rv
o Generating Functions
Poisson approximation to the Binomial
Čebyšëv (Чебышёв) Inequality
Law of large numbers ( weak and strong formulation )
Continuous random variables with values in R^n
o Absolutely continuous rv
o Density Function (continuous)
o Joint (continuous) laws, marginals and conditional independence
o Examples: uniform, exponential, Gaussian, Gamma, etc.
o Mean, variance, covariance operators
o Normal laws
o Transformations of rv in R^n
o Conditional expectation (as a rv)
o Characteristic functions
o Moments of a rv
Convergence
o rv theory in the measure theory framework
o various types of convergence for sequences of rv
o central limit theorem and the Gaussian approximation
Examination Methods
Written exam