Introduction to Probability, Statistics and Operations Research


The main objectives are: a) to introduce students to the basic notions on statistics and probability. The students will be prepared to easily handle the requirements of a professional activity, concerning probabilities and statistics; b) introduce concepts from a few areas of Operations Research, namely Linear Programming, Integer Programming, Project Management and Decision Theory.


General characterization





Responsible teacher

Filipe José Gonçalves Pereira Marques, Manuel Valdemar Cabral Vieira


Weekly - 5

Total - 70

Teaching language



Elementar knowledge about Mathematical Analysis, pointing out: elementar sets algebra, limit of sequences, primitives, integrals and real functions of one or more real variables. Elementar knowledge about Linear Algebra, namely, matrices, systems of linear equations and vector spaces.


Part I

1. Mood, A. M., Graybill, F. A. e Boes, D. C. (1974). Introduction to the Theory of Statistics, 3ªed. McGraw-Hill, New York

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

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

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

Part II

1. Wayne Winston, Operations Research: Applications and Algorithms, Duxbury Press; 4th. Edition, 2003

2. Introdução à Programação Linear, J.O. Cerdeira, texto de apoio à unidade curricular Introdução às Probabilidades e Estatística e Investigação Operacional, 2013

Teaching method

Classes operate on a theoretical and practical regime.

In classes are exposed the theoretical conceptssome demonstrations are carried out and simultaneously its application are illustrated through examples and exercises.

Pupils have supporting texts on all matter including exercises and application problems.

Substantial part of the study is done on learner autonomywith the aid of notes and bibliographic other mediaand with the support of teachers to answer questions at pre-established times.

Evaluation method

The u.c. consists of two modules: Probability and Statistics (PS) and Operational Research (OR)​​, which will be evaluated independently, each one for a maximum value of 10 points. 

Attendance to the UC is obtained with at least 3/5 of attendance in practical classes (P) taught in each moduleStudents with special status or who achieved a grade of Sufficient in the previous academic year are exempt from attending classes.

Continuous Evaluation

The PS contents taught are evaluated in the 1st Test that will be held on a date to be posted by the FCT-UNL. Let N1 be the grade obtained by the student.

The IO contents taught are evaluated in the 2nd Test that will be held on a date to be posted by FCT-UNL. Let N2 be the grade obtained by the student.

The student is approved if N1+N2>=9.5. The final grade will result from the rounding to units of N1+N2.

Students with a grade N1+N2>=18 may be called to perform an additional examination. Students in such conditions that fail this additional examination will be evaluated with a final grade of 18 points.


The final exam will be held on a date to be announced by FCT-UNL.

Students who have not been approved during the continuous evalutaion or that, having passed the UC, wish to improve their classification, may take the final exam.

Prior to the exam, the student must indicate whether he/she wishes to be evaluated in the two modules (full exam) or just in one of the modules. In the latter case, the new module classification replaces the old one and the final grade is recomputed accordingly.

Again, the student passes the course if the grade in the exam is >=9.5.

Students who intend to take the exam to improve the grade obtained in continuous evaluation must sign up for this exame in the academic services or CLIP. In this case students will have to take the full exam. The student’s final course grade will be the maximum between the grade obtained in the exam and the one obtained via continuous evaluation.

Students with a grade in the exam  >=18 may be called to perform an additional examination. Students in such conditions that fail this additional examination will be evaluated with a final grade of 18 points. 


During any evaluation students are only allowed to use a scientific calculator.

Subject matter

Part I - Probability and Statistics.

1 - Basic notions of Probability.

2 - Random variables.  

3 - Moments of random variables.

4 - Some important distributions. Central Limit Theorem.

5 -  Point and interval estimation.

6 - Hypothesis testing.

7 - Simple linear regression

8 - R Introduction

Part II - Operational Research:

1 -  Linear Programming:

      1.1 - Formulations of Linear Programming problems.  

       1.2 - Graphical resolution.

       1.3 - The Simplex method. Artificial Variable Technique. 


2 - Integer Programming:
         2.1 - Formulations of Integer Programming problems.

         2.2 - Methods for solving Integer Programming problems.

 3 - Project Management:

       3.1 - Critical Path Method.

       3.2 - PERT technique.

       3.3 - Construction of the Time Chart and Resource Leveling.

       3.4 - Reduction of the project duration.

4- Decision Theory:

       4.1 -  Decisions under risk and under uncertainty.

       4.2 - Decision Trees.