Probability and Statistics D


It is objective of this subject to teach the basics about the theory of probability, namely about probability, conditional probability, independence, random variables - their distribution, moments and some other characteristics - and the central limit theorem.


The above matters are then used to teach the fundamentals about statistics, as the notion of population, sample and random sample, estimators, their sample distributions and some other properties, point estimation, estimation by confidence interval, hypotheses testing and simple linear regression.


The key important point here is one of teaching these subjects in a way that, in the future, students can: use adequately these probabilitiy and statistical tools, judiciously analyse statistical results and easily learn other statistical methods (not included in the discipline syllabous).


General characterization





Responsible teacher

Maria de Fátima Varregoso Miguens


Weekly - 4

Total - 68

Teaching language



Basics of mathematical analysis, pointing out: some topological notions; analyses, diffrerential and integral knowledges about real (or R2) functions with one or more real variables.


Guimarães, R.C. e Cabral, J.S. (1997). Estatística, McGraw-Hill

Pedrosa, A. (2004). Introdução Computacional à Probabilidade e Estatística, Porto Editora

Murteira, B., Ribeiro, C.S., Silva, J.A. e Pimenta, C. (2002). Introdução à Estatística, McGraw-Hill

Montgomery e Runger (2002). Applied Statistics and Probability for Engineers. Wiley

Miguens, M.F.V. (2019). Textos de Apoio às disciplinas de serviço do DM. para a área de Probabilidades e Estatística. DMAT

Rohatgi (1976). An Introduction to Probability Theory and Mathematical Statistics. Wiley

Sokal e Rohlf (1995). Biometry. Freeman

Tiago de Oliveira (1990). Probabilidades e Estatística: Conceitos, Métodos e Aplicações, vol. I, II. McGraw-Hill

Paulino e Branco (2005). Exercícios de Probabilidade e Estatística. Escolar Editora

Robalo, A. (1994). Estatística Exercícios, Vol I, II. Edições Sílabo

Teaching method

Lectures and problem-solving sessions, with wide participation of students.

Evaluation method

Frequency: Obtained with at least two thirds of attendance in classes taught in each module.

The students obtain approval if the weighted average of the three tests is greater than or equal to 9.5. If a student does not attend a test, this test will come with the factor of "0 x corresponding percentage" for the final classification.

Final mark = 40%T1 + 40%T2 + 20%T3

The evaluation by exam is valid both for grade improvement as for discipline approval. The student with a final score greater than or equal to 17.5 should carry out an oral defense of note. Otherwise, will get a final score of 17.0.

More detailed rules are available in the Portuguese version

Subject matter

Short syllabus

1. Basic notions of probability: Probability function and probability calculus of probabilities. Conditional probability (Bayes theorem) and independence of events

2. Discrete random variables (r.a.): Probability distributions and moments 

3. Discrete random vectors: Joint and marginal distribution functions: Independence of v.a.''''''''s; Moments (correlation coefficient; Moments properties for linear tranformations of r.a.''''''''s 

4. Continuous random variable: Density probability function, calculus of probabilities and moments

5 . Some important discrete distributions: Hipergeimetric, Bernoulli and Binomial, Poisson and his relation to Binomial, Poisson Process and Geometric.

6. Some important continuos distributions: Uniforme, Exponencial (and relations to Poisson Process), Normal (special emphasis) , t-Student and Qui-Square.

7. Central Limit Theorem 

8. Basic notions of statistics, Random sample (r.a.) and stochastic properties for a resamplig extraction sample.

9. Pontual estimation: Desirable properties of  no bias, efficiency and consistency

10. Interval estimation (Pivotal Method)

11. Hypothesis testing: Elementar concepts and their implementation for population parameters such as mean value, variance and porportion.

12. Simple linear regression and exemples for linearizable models.



Programs where the course is taught: