Probability and Statistics I


Give the students a good background in probability so that they will be able to easily relate with the most common probability distributions of both categorical and continuous random variables, allowing them to better understand the meaning of some of the more important results pertaining operations on probabilities and on random variables, namely sums of these latter ones, in order to be possible to build a solid background for a correct use of the more basic inferential procedures, namely the ones on proportions, means, variances, quantiles and medians.

General characterization





Responsible teacher

Carlos Manuel Agra Coelho


Weekly - 5

Total - 70

Teaching language



The students should be provided with basic knowledge about calculus (mathematical analysis: geometric and arithmetic progressions, sumations, series, derivation and integration)


Coelho, C. A. (2008). Tópicos em Probabilidades e Estatística, Vol. I, Vol. II (Cap.s 6,7).

Mood, A. M., Graybill, F. A. e Boes, D. C. (1974). Introduction to the Theory of Statistics, 3ª ed., J. Wiley & Sons, New York.

Montgomery, D. C. e Runger, G. C. (1998). Applied Statistics and Probability for Engineers, 2ª ed., J. Wiley & Sons, New York.

Ross, S. M. (1999). Introduction to Probability and Statistics for Engineers and Scientists. J Wiley & Sons, New York.

Murteira, B. J. F. (1990). Probabilidades e Estatística, Vol I, 2ª ed., McGraw-Hill Portugal, Lisboa.

Rohatgi, V. K. (1976). An Introduction to Probability Theory and Mathematical Statistics. J. Wiley & Sons, New York.


Teaching method

  • 2 weekly Theoretical Classes (in a total of 3 hours per week), where the concepts are introduced and analysed and the main results are derived and proven. Illustrative examples are also shown.
  • 1 weekly Lab of 2 hours where exercises and problems pertaining the concepts and results shown in the Thoeretical classes are solved.

Evaluation method

Evaluation during the 2nd Semester of the school year 2019/2020

Students of the Curricular Unit Probabilidades e Estatística I will have in the second semester of the school year 2019/2020 will have at their disposal 4 different modalities of evaluation, to obtain approval for this Curricular Unit. In any of these 4 modalities each student will have to obtain a final grade equal or greater than 9.5 (on a 0-20 scale) in order to obtain approval, and the students with a final grade equal or larger than 18 will have to be subject to na oral examination, without which the final grade will be 17 (this oral examination will be, at the present semester done online and the final grade can be anything between 17 and the original grade).

The 4 modalities of evaluation are:

1 – only final Exam

2 – only 2 Tests, situation in which the 1st Test will have a weight of 40% and the 2nd Test a weight of 60% for the final classification

3 – 2 Tests and final Exam, situation in which the 1st Test will have a weight of 20%, the 2nd a weight of 25% and the Exam a weight of 55%, being the final grade the best between the grade obtained with these ponderations and the grade from the Exam (all grades on a 0-20 scale)

4 – only 1 Test and final Exam, in which case the Exam will have a weight equal to 100% minus the weight of the Test that was taken by the student (20% for the 1st Test or 25% for the second Test), being the final grade the best grade between the Exam grade and the grade obtained with the above ponderation (all grades on a 0-20 scale).

In all online Tests and Exams, and according to the recent recommendations from UNL, all students will be required to have their video on during the time that the Test or the Exam will be on, which will be done through the Zoom system, using a link given by the person in charge of the Curricular Unit. All submissions of the students responses can only be done through the Moodle platform of FCT/UNL.

 1. Pre-Requisites

In order to be able to have access to the course evaluation, both to midterms and tests and also to the Exam, students on a first enrolment need to have the presence in at least 80% of both Labs and Classes, being this percentage reduced to 2/3 of both Labs and Classes for the other students (once obtained, this presence score, in case it will be necessary, wil remain valid for the following year).

 2. Evaluation

  • The recommended form of evaluation consists in 2 Tests:
    • 1st Test - weight: 40% -  April 8 (allowed and necessary the use of a simple calculator, which cannot be a graphic one)
    • 2nd Test - weight: 60% -  June 12 (no calculator allowed)
    • The student who has an average grade of at least 9.5 (on a 0-20 scale) will be approved in the course.
  • Students who obtained a final grade from tests less than 9.5 (on a 0-20 scale), may have access to a final Exam, in case they have attended at least 2/3 of Labs and 2/3 of Classes.
  • Also the students who had a grade equal or greater than 9.5 from tests may have access to the Final Exam in order to improve their grade.
  • Students with a final grade of more than 17 (on a 0-20 scale) have to go through an oral examination, or their final grade will be equal to 17.

Subject matter

1 – Combinatorics (short review) 

2 – Elementary Probability Theory


  • Random experiment and Outcome space
  • Event and Event Space
  • The concept of Probability. Probabilities Properties
  • Conditional Probability and independence of events
    • Some useful and iteresting results on Conditional Probabilities
    • Conditional Independence and (Marginal) Independence
    • Odds and Odds ratio
    • Illustrative example of the reason of the definition of the Independence of 3 or more events
    • Examples of application (of the notion of conditional probability, Bayes formula and Bayes and Total Probability Theorems)
    • The Borel-Cantelli Lemmas

3 – Random variables and Probability Distributions

  • Definition of random variable. Examples
  • Probability Distribution functions. Properties. Quantiles
  • The Survival and Risk functions – two alternative ways of representing the distribution of a r.v.
    • Development and study of a Risk function
    • Survival and Risk functions for discrete r.v.''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s
  • Expected value. Properties. Moments. Some important inequalities involving moments
  • Moment generating functions and characteristic functions
  • The distribution of Y = g(X)

4 – Joint and conditional distributions of random variables

  • Joint distribution of two or more random variables
  • Joint and marginal moments
  • The joint moment generating function
  • Conditional distributions and independence
    • Conditional moments
    • The conditional expected value
    • Some additional notes on the conditional expected value
    • Independence of r.v.s
    • Consequences of independence
    • Other conditional distributions
    • Truncated distributions as conditional distributions
  • Joint distributions of r.v.s of different types
  • The distribution of (Y1, Y2) = g(X1,X2)
  • The distributions of Sum, Difference, Product and Ratio of two r.v.s
  • Mixtures

5 – Discrete random variables

  • The Uniform distribution
  • The Geometric distribution
  • The Negative Binomial distribution
  • The Bernoulli distribution
  • The Binomial distribution
  • The Hipergeometric distribution
  • The Poisson distribution

6 – Continuous random variables

  • The Exponential distribution
  • The Normal distribution
  • The chi-square distribution
  • The T distribution
  • The F distribution
  • The Gamma distribution

7 – Brief reference to multivariate distributions

  • The Multinomial distribution
  • The Multivariate Normal distribution



Programs where the course is taught: