Universidade Nova de Lisboa

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Faculdade de Ciências e Tecnologia

Probability and Statistics

Code

3645

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

João Tiago Praça Nunes Mexia

Weekly hours

5

Total hours

84

Teaching language

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.

Prerequisites

Reasonable knowledge about real differentiation and integration.

Subject matter

 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
  • Discrete random variable
  • Probability function
  • Mean value and properties
  • Variance, standard variation and properties
  • Continuous random variable
  • density function
  • Distribution 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, Exponencial, Weibull and Normal
  • Central Limit Theorem

    • STATISTIC

  • 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
  • Methods of point estimators: Method of moments
  • Tests of hypotheses
  • Elementar concepts
  • Hypothesis, null hypothesis, alternative hypothesis, simple and compound hypotheses
  • Decision and critical region
  • Decision errors and probabilites
  • Significance level and P-value
  • Bilateral and unilateral tests for the: mean value, variance, standard deviation, proportion, fifference of mean values, ratio of variances
  • Tests for validation on population conditions
  • Randomeness test
  • Testing for godness of fit to normality: chi-square test
  • Confidence interval estimation: elementar concepts
  • Confidence interval estimation for the: mean value, variance, standard deviation, proportion, fifference of mean values, ratio of variances
  • 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
  • Contingency tables: test for independence

Bibliography

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.Guimarães, R.C. & Cabral, J.A.S. (1997),

Estatística, 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.

Teaching method

Lectures and problem-solving sessions, with wide participation of students and informatic software.

Evaluation method

Evaluation

The evaluation is one of two: either the students take 2 tests or one exam. Each test covers half of the subjects taught, students must be classified over 7/20 in each test and they are approved in this subject if the average grade of the 2 tests is 10/20 or more or they get a grade of 10/20 or more in the exam. Students may have to do an oral examination to confirm a grade of more than 18/20.

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