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.

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

Written exam