Learning objectives
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.
Prerequisites
Funzioni di una variabile B
Course unit content
<strong>The main part is in common with the course <em>Statistica </em>of Facoltà di Ingegneria:</strong><br />
<br />
Elementary combinatorics.<br />
Probability spaces, conditioning, independence, total probabilities and Bayes formulae.<br />
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.<br />
Common types of random variables (Bernoulli, binomial, Poisson, hypergeometric, uniform, exponential, Gaussian, chi-square, gamma and t).<br />
Convergence in probability, law of large numbers, central limit theorem, continuity correction.<br />
Populations, samples, descriptive statistics, estimators (bias and consistency), sample mean and sample variance.<br />
Parametric confidence intervals (gaussian, Bernoulli and exponential populations).<br />
Nonbayesian parametric tests, bi- and unilateral (same populations as above), tests for comparing two gaussian populations.<br />
<br />
<strong>The enhaced part is only devoted to the students of Facoltà di Scienze.</strong><br />
<br />
Properties of binomial and multinomial coefficients.<br />
Abstract probability spaces. Sigma-fields. Sigma-field generated by a set. Borel sets of R. Carathéodory theorem with uniqueness (no proof).<br />
Asymptotic events (liminf and limsup), Fatou lemma, first Borel-Cantelli lemma. Strong law of large numbers for a sequence of coin tosses.<br />
Abstract random variables. Links between the law and the cumulative function. Skorokhod theorem. Abstract expected value.<br />
Complex examples of conditioned laws. Bernoulli process. Geometric and binomial negative law.
Bibliography
<ol>
<li>S. Ross - Probabilità e statistica per l'ingegneria e le scienze - Apogeo 2003 (the english version is ok too)</li>
<li>D. Williams - Probability with martingales - Cambridge University Press 1991</li>
</ol>
