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.
Discrete probability spaces. Elements of combinatorial calculus. Conditional probability and independence.
Applications: random permutations, percolation.
Discrete random variables and distributions. Independence of random variables. Expectation and inequalities. Notable classes of discrete random variables.
Applications: the law of small numbers, the binomial model in finance, the collector's problem.
Probability spaces and general random variables.
Absolutely continuous random variables. Notable classes of absolutely continuous random variables. Absolutely continuous random vectors. The Poisson process. Normal laws.
The law of large numbers. The central limit theorem and normal approximation.
Elements of stochastic simulation.
Textbook: F. Caravenna, P. Dai Pra, Probabilità. Un'introduzione attraverso modelli e applicazioni - UNITEXT - La matematica per il 3+2. Springer-Verlag, 2013.
|Francesca Caravenna, Paolo Dai Pra||Probabiltà - Un primo corso attraverso modelli e applicazioni (Edizione 1)||Springer-Verlag||2013|
Written exam, with exercises and theoretical questions.
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