Probabilities

Recommended Prerequisites

Infinitesimal Analisys I, II, III.

Teaching Methods

Lectures are expository, but also include examples and exercises to help grasping theory and gaining practical experience about the acquired knowledge. During the semester, students can also use tutorial time to discuss their ideas and clarify possible understanding difficulties.

Learning Outcomes

The development of this curricular unit takes into account the main guidelines of a basic course in Probability: to describe, analyze and interpret real-world situations through nondeterministic mathematical models and to provide the essential theoretical skills for subsequent studies in areas as Statistics or Stochastic Processes.

This course allows developing the following skills: ability to calculate; knowledge of mathematical results; ability to generalize and abstract; formulating and solving problems; design and use of mathematical models for real situations; writing and oral expression rigorous and clear. On a personal level, it allows to develop individual initiative, teamwork, research and independent learning.

Work Placement(s)

No

Syllabus

Chapter1 Introduction to probability theory. Basic probability theory: classical and frequentist approaches. Probability spaces. Conditional probabilities. Independence

Chapter2 Probability distributions on IR. Real random variable. Discrete probability distributions, absolutely continuous probability distributions and mixed probability distributions. Location and dispersion parameters: moments and quantiles

Chapter3 Probability distributions on IRn. Real random vectors. Discrete probability distributions, absolutely continuous probability distributions. Independent real random variables. Moments of real random vectors: mathematical expectation, variances-covariances matrix.

Chapter4 Characteristic function and random variables independence.  Characteristic function and moments

Chapter5 Stochastic Convergences. Convergence in distribution. Almost sure convergence, convergence in probability and mean square convergence. Law of large numbers and central limit theorem.

Head Lecturer(s)

Maria de Nazaré Simões Quadros Mendes Lopes

Assessment Methods

Continuous Assessment
Mini Tests: 30.0%
Frequency: 70.0%

Final Assessment
Exam: 100.0%

Bibliography

E. Gonçalves, N. Mendes-Lopes, Probabilidades - Princípios Teóricos, 2ªed., Escolar Editora, 2013

 

K. I. Chung, A Course in Probability Theory and Mathematical Statistics. Academic Press, 1974

 

D. Foata and A. Fuchs, Calcul des probabilités, Masson, 1996

 

D. Pestana and S. Velosa, Introdução à Probabilidade e à Estatística, Vol. I, 2ª ed., Fundação Calouste Gulbenkian, 2006

 

S. Resnick, A probability path, I. Birkhauser, 1999

 

Ph. Tassi and S. Legait, Théorie des Probabilités en vue des Applications Statistiques, Éditions Technip, 1990