## Courses Catalogue

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Our catalogue provides a description of the courses offered at NOVA as well as useful information about our Schools.

# Probability and Statistics

3645

### Department

Departamento de Matemática

6.0

### Teacher in charge

Carlos Manuel Agra Coelho, Filipe José Gonçalves Pereira Marques

5

84

Português

### Objectives

Intuitive comprehension of the most important concepts of probabilities, such as: random variables; distributions; moments and Central Limit Theorem. It is stressed the importance of those mathematical tools in science and technology.

Comprehension of the classic statistical methods and concepts, such as; population and sample; estimators and pontual estimation; sampling distributions; confidence estimation; hypotheses testing; fit testing and regresion analysis.

# PROBABILITY

• Random experiment; Sample space; Random event; Algebra of events
• Axioms of probability e 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
• # STATISTICS

• Elementar concepts in statistic
• Population, random sample and observed sample
• Simple random sample
• Pontual estimation
• 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
• Elementar concepts
• Hypothesis, null hypothesis, alternative hypothesis, simple and compound hypotheses
• Decision and critical region
• Decision errors and probabilites
• Significance level
• 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
• Randomeness 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

### Bibliography

Guimarães, R.C. & Cabral, J.A.S. (2007), Estatística, 2th ed., McGraw-Hill, Portugal.

Pedrosa, A.C.& Gama, S.M.A. (2004), Introdução Computacional à Probabilidade e Estatística, Porto Editora, Porto.

Montgomery, D.C.& Hines, W.W. (1990), Probability and Statistics in Engineering and Management Science, 3rd ed., John Wiley & Sons, New York.

Montgomery, D.C. & Runger, G.C. (1999),
Applied Statistics and Probability for Engineers, 2nd ed., John Wiley

Larson, H.J. (1969), Introduction to Probability Theory and Statistical Inference, 2nd ed., John Wiley & Sons, New York.

Mood, A.M., Graybill, F.A. & Boes, D.C. (1974),
Introduction to the Theory of Statistics, 3rd ed., McGraw-Hill, Singapore.

Pestana, D.D. & Sílvio Filipe Velosa, S.F. (2002) Introdução à Probabilidade e à Estatística, vol. I, Fundação Calouste Gulbenkian, Lisboa.

Tiago de Oliveira, J. (1990),
Probabilidades e Estatística: Conceitos, Métodos e Aplicações, vol. I, II, McGraw-Hill, Portugal.

Murteira, B.J.F. (1990), Probabilidades e Estatística, vol. I, II, McGraw-Hill, Portugal.

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

Robalo, A. (1994), Estatística - Exercícios, vol. I, II, Edições Sílabo, Portugal.

### Evaluation method

Evaluation

To be accepted in tests/final exam evaluation, students must attend, at least, two thirds of the lectures.

The evaluation is one of two: either the students take 2 tests or one exam. The tests and the exam are rated on a scale of 0 to 20. Each test covers half of the subjects taught, students must be classified over 8.0 (exact value)  in each test and they are approved in this subject if the average grade of the 2 tests is 10 (that is 9.5) or more, or if they get a grade of 10 (that is 9.5) or more in the exam.

Students may have to do an oral examination to confirm a grade of more than 18 (that is 17.5).

Students must register in the CLIP webpage in order to do the tests or exams.

More detailed rules are available in the Portuguese version.