ELEMENTS OF PROBABILITY
Learning outcomes of the course unit
This is an introductory course on probability theory. There is a quick glance of the abstract-axiomatic setting and of measure theory, and a thorough study of laws of real random variables, continuous and discrete. Some elementary arguments of statistics are included, with a view towards applications and problem solving.
Funzioni di una variabile B
Course contents summary
The main part is in common with the course Statistica of Facoltà di Ingegneria:
Probability spaces, conditioning, independence, total probabilities and Bayes formulae.
Continuous and discrete random variables, distribution functions (cumulative, density, mass), joint distributions, transformations. Expected value, variance, median, mode. Min, max and sum of independent random variables.
Common types of random variables (Bernoulli, binomial, Poisson, hypergeometric, uniform, exponential, Gaussian, chi-square, gamma and t).
Convergence in probability, law of large numbers, central limit theorem, continuity correction.
Populations, samples, descriptive statistics, estimators (bias and consistency), sample mean and sample variance.
Parametric confidence intervals (gaussian, Bernoulli and exponential populations).
Nonbayesian parametric tests, bi- and unilateral (same populations as above), tests for comparing two gaussian populations.
The enhaced part is only devoted to the students of Facoltà di Scienze.
Properties of binomial and multinomial coefficients.
Abstract probability spaces. Sigma-fields. Sigma-field generated by a set. Borel sets of R. Carathéodory theorem with uniqueness (no proof).
Asymptotic events (liminf and limsup), Fatou lemma, first Borel-Cantelli lemma. Strong law of large numbers for a sequence of coin tosses.
Abstract random variables. Links between the law and the cumulative function. Skorokhod theorem. Abstract expected value.
Complex examples of conditioned laws. Bernoulli process. Geometric and binomial negative law.
- S. Ross - Probabilità e statistica per l'ingegneria e le scienze - Apogeo 2003 (the english version is ok too)
- D. Williams - Probability with martingales - Cambridge University Press 1991