Elements of Probability and Number Theory

Objectives

The primary aim of this course is to provide students with basic knowledge and skills in Probability Theory and Number Theory. As a consequence, the learning outcomes of the curricular unit are the following:

  • Handle axiomatic theory, conditioning and probability trees.
  • Deal with parametric distributions relevant for applications, as well as approximations and convergence.
  • Apply the concepts of parameter estimation and modeling to real datasets.
  • Use the euclidean algorithm to evaluate the greatest common divisor of two integers
  • Solve diaphantine equations
  • Solve linear congruences and systems of linear congruences

Program

I. Basic notions of Probability Theory

  • Probability: axioms, conditioning and independence.
  • Probability distributions, moments, independence, approximations and stochastic convergence.
  • Parametric models, parametric estimation and data modeling. II. Basic notions of Number Theory
  • Divisibility; greatest common divisor of two integers; the euclidean algorithm.
  • Prime numbers; the Fundamental Theorem of Arithmetic.
  • Diaphantine equations.
  • Linear congruences and systems of linear congruences.

Bibliography

  • Pestana, D. D. e Velosa, S. F. (2010). Introdução à Probabilidade e à Estatística, Vol. I (4a ed.). Fundação Calouste Gulbenkian.
  • Forsyth, D. (2018). Probability and Statistics for Computer Science. Springer
  • Ross, S (2002). Probability Models for Computer Science. Harcourt / Academic Press.
  • Prügel-Bennett, A. (2020). The Probability Companion for Engineering and Computer Science. Cambridge University Press.
  • Jones, G. A. and Jones, J. M. (2005). Elementary Number Theory, Springer Undergraduate Mathematics Series, 8th printing, London
  • Burton, D. (2010). Elementary Number Theory, McGraw-Hill Education, 7 edition

Updated: