Probability and Statistics I
Objectives
General characterization
Code
10975
Credits
6.0
Responsible teacher
Carlos Manuel Agra Coelho
Hours
Weekly  5
Total  70
Teaching language
Português
Prerequisites
The students should be provided with basic knowledge about calculus (mathematical analysis: geometric and arithmetic progressions, sumations, series, derivation and integration)
Bibliography
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., McGrawHill 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 020 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 020 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 020 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. PreRequisites
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 020 scale) will be approved in the course.
 Students who obtained a final grade from tests less than 9.5 (on a 020 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 020 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 BorelCantelli 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 chisquare distribution
 The T distribution
 The F distribution
 The Gamma distribution
7 – Brief reference to multivariate distributions
 The Multinomial distribution
 The Multivariate Normal distribution