## Courses Catalogue

We welcome you to explore NOVA’s academic offerings.
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

João Tiago Praça Nunes Mexia

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.

### Prerequisites

Reasonable knowledge about real differentiation and integration.

# 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.