Faculdade de Ciências e Tecnologia

Probability and Statistics



Academic unit

Faculdade de Ciências e Tecnologia


Departamento de Matemática



Teacher in charge

Luís Pedro Carneiro Ramos

Weekly hours


Total hours


Teaching language



The aim of the course is to provide students a basic knowledge of Probabilities and Statistics which are an indispensable tool for decision making under uncertainty, present in many areas in the field of Engineering. In addition, the course helps students gain an appreciation for the diverse applications of statistics and its relevance to their lives and fields of study.

At the end of the unit students will have acquired skills that enable them:

-Know and understand the basic elements of the theory and the calculus of probabilities

-Describe the main probabilistic distributions of discrete and continuous variables and applies them in the description of random phenomena

-Infer about population parameters based on sample distributions

-Build-statistical models, which establish a functional relationship between variables


Basics of mathematical analysis, pointing out some topological notions, primitives, integrals and functions of more than one variable.

Subject matter


    • Random experiment; Sample space; Random event; Algebra of events
    • Axioms of probability and  addition rules
    • Conditional probability
    • Total probability rule
    • Bayes theorem
    • Random variable
    • Distribution function
    • Discrete random variable
    • Probability function
    • Mean value and properties
    • Variance, standard variation and properties
    • Continuous random variable
    • density function
    • Mean value, variance and standard deviation
    • Chebychev inequality
    • Discrete random pair
    • Joint and marginal probability functions
    • Covariance and properties
    • Correlation coefficient and properties
    • Important discrete distributions: Hipergeometric, Binomial, Poisson
    • Important continuous distributions: Uniform, Exponential and Normal
    • Central Limit Theorem



    • Elementar concepts in statistic
    • Population, random sample and observed sample
    • Simple random sample
    • Pontual estimation
    • Estimators and estimatives
    • Desirable properties for estimators: unbiased and minimum variance estimators
    • Method of moments


  • Confidence interval estimation: elementar concepts
  • Confidence interval estimation for the: mean value, variance, standard deviation, proportion, difference of mean values, ratio of variances


  • Tests of hypotheses
  • Elementar concepts
  • Hypothesis, null hypothesis, alternative hypothesis, simple and compound hypotheses
  • Decision and critical region
  • Decision errors and probabilites
  • Significance level
  • Bilateral and unilateral tests for the: mean value, variance, standard deviation, proportion, difference of mean values, ratio of variances
  • Testing for godness of fit to normality: chi-square test
  • Randomeness test
  • Contingency tables: test for independence
  • Simple linear regression
  • Pontual and confidence interval estimation for the model parameters
  • Bilateral and unilateral tests for the model parameters
  • Testing the quality of the model
  • Pontual and confidence interval estimation on the: mean response and new observation prediction




Guimarães, R.C. & Cabral, J.A.S. (2007), Estatística, McGraw-Hill.

Montgomery, D.C. & Runger, G.C. (2011), Applied Statistics and Probability for Engineers, John Wiley.

Pedrosa, A.C.& Gama, S.M.A. (2004), Introdução Computacional à Probabilidade e Estatística, Porto Editora.

Pestana, D.D. &  Velosa, S.F. (2002) Introdução à Probabilidade e à Estatística, Fundação Calouste Gulbenkian, Lisboa.

Robalo, A. (1994), Estatística - Exercícios, vol. I, II, Edições Sílabo, Portugal.

Teaching method

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

Evaluation method

The classes take place with an oral presentation of the subjects, followed by practical examples. Learning is complemented by solving exercises.

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 = 30%T1 + 50%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 18.5 should carry out an oral defense of note. Otherwise, will get a final score of 18.0.

 More detailed rules are available in the Portuguese version.