Probability and Statistics
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
Frederico Almeida Gião Gonçalves Caeiro
Weekly - 5
Total - 78
Basics of mathematical analysis, pointing out some topological notions, primitives, integrals and functions of more than one variable.
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.
Lectures and problem-solving sessions, with wide participation of students.
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 two 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 = 0.5 * T1 + 0.5 * T2
EVALUATION BY EXAM
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 17.5 should carry out an oral defense of note. Otherwise, will get a final score of 17.0.
More detailed rules are available in the Portuguese version.
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
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
Hypothesis, null hypothesis, alternative hypothesis, simple and compound hypotheses
Decision and critical region
Decision errors and probabilites
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
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