Teoria da Probabilidade

Objectives

The goal of this course consists in exposing the students to probability theory since the basic notion of random variable, passing through the concepts of convergence of sequence of random variables up to the theory of martingales in discrete time.

Syllabus

Brief recap on probability spaces, random variables and random vectors, stochastic independence, mathematical expectation. Convergence of sequences of random variables: different types of convergence and relationship between them. Uniform integrability and moments. Borel-Cantelli lemmas. Convergence of series of random variables, Kolmogorov's theorems, equivalent sequences. Laws of large numbers: weak law and strong laws. Characteristic functions: properties, uniqueness theorem, inversion formula and its relationship with weak convergence, central limit theorem. Conditional expectation: basic properties, conditional expectation with respect to a finite number of random variables and with respect to sigma algebras. Theory of discrete time martingales: properties, upcrossing, convergence theorems, stopping times, optional stopping theorems, aplications.

Prerequisites

Differential and Integral Calculus I, II and III, Probability and Statistics and Complements of Probability, or equivalent.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Laboratorial Component

Does not have.

Programming And Computing Component

Does not have.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.