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

Isabel Cristina Silva Correia, Vanda Marisa da Rosa Milheiro Lourenço


Weekly - 6

Total - 86

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

Available soon

Evaluation method

Evaluation rules

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: Obtained with at least 2/3 of class attendance for each module. Students with special status or that obtained course attendance in the previous year are exempted from this year''s course attendance.


Module PS

1st test: 10 points (N1)

Module OR

2st test: 10 points (N2)

The first and second tests concern PS and OR and will be graded to 10 points. The final classification will be equal to N1+N2.

A student passes the course if he/she complies with the following 4 requirements:

1) he/she has achieved course attendance or is exempted from it

2) N1>=4

3) N2>=4

4) N1+N2 >=10

By final exam:
Only students who achieved attendance (or who are exempted from it) are elligible for the final exam. Prior to the exam (usually 1-2 weeks before), 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 he/she has achieved course attendance, the sum N1+N2 >=10 and as long as both N1 and N2 are >=4.

Improving your course grade:

Those student''s who have already passed the course are elligle for course grade improvement via the final exam, in which case they must sign up beforehand in the "Serviços académicos" or CLIP. In addition, students should be aware that for course grade improvement both modules need to be assessed and therefore they should prepare themselves accordingly.

During course/modules assessment, students are 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.