Learning outcomes

The course introduces basic concepts of probability theory, with particular emphasis on its formal description starting from its axiomatization due to A. Kolmogorov.

The course aims to provide the notions needed in order to understand and apply in complete autonomy the theory that lies behind probability in various problems of both physics and daily life.

No special notions will be required, the student must have learned the mathematical methodology on which the first year of the bachelor degree is based.

Program

1) Probability spaces: introduction to different notions of probability, probability axiomatization, recalls of measure theory, sample space and events, first consequences of the axioms of probability, conditional probability, Bayes theorem and total probability theorem, independence of events;

2) Discrete random variables: definition and motivation of the notion of random variable, discrete random variables, mean and variance of random variables and functions of random variable, notable random variables and their properties: Bernoulli, binomial, Poisson, geometric and hypergeometric, joint laws and covariance;

3) Continuous random variables: definition of (absolutely) continuous random variables, mean, variance and moments of continuous random variable, notable random variables and their properties: uniform, normal, exponential, Gamma, Beta, Cauchy and Maxwell-Boltzmann , joint density function, conditional expectation and multivariate Gaussian laws;

4) Convergence and approximation: Markov and Chebyshev's inequality, law (weak and strong) of large numbers, convergence in law and probability, central limit theorem.

Examination Methods

The final exam consists of a written exam followed, in case the written exam is passed, by an oral examination .