# Probability and Statistics D

### Code

10354

### Academic unit

Faculdade de Ciências e Tecnologia

### Department

Departamento de Matemática

### Credits

6.0

### Teacher in charge

Maria de Fátima Varregoso Miguens

### Weekly hours

4

### Total hours

56

### Teaching language

Português

### Objectives

It is objective of this subject to teach the basics about the theory of probability, namely about probability, conditional probability, independence, random variables - their distribution, moments and some other characteristics - and the central limit theorem.

The above matters are then used to teach the fundamentals about statistics, as the notion of population, sample and random sample, estimators, their sample distributions and some other properties, point estimation, estimation by confidence interval, hypotheses testing and simple linear regression.

The key important point here is one of teaching these subjects in a way that, in the future, students can: use adequately these probabilitiy and statistical tools, judiciously analyse statistical results and easily learn other statistical methods (not included in the discipline syllabous).

### Prerequisites

Basics of mathematical analysis, pointing out: some topological notions; analyses, diffrerential and integral knowledges about real (or R2) functions with one or more real variables.

### Subject matter

**Short syllabus**

1. Basic notions of probability: Probability function and probability calculus of probabilities. Conditional probability (Bayes theorem) and independence of events

2. Discrete random variables (r.a.): Probability distributions and moments

3. Discrete random vectors: Joint and marginal distribution functions: Independence of v.a.''s; Moments (correlation coefficient; Moments properties for linear tranformations of r.a.''s

4. Continuous random variable: Density probability function, calculus of probabilities and moments

5 . Some important discrete and continuous distributions . Special emphasis on the Normal distribution

6. Central Limit Theorem

7. Basic notions of statistics, Random sample (r.a.) and stochastic properties for a resamplig extraction sample.

8. Pontual estimation: Desirable properties of no bias, efficiency and consistency

9. Interval estimation (Pivotal Method)

10. Hypothesis testing: Elementar concepts and their implementation for population parameters such as mean value, variance and porportion.

11. Simple linear regression

### Bibliography

Guimarães e Cabral (1997). *Estatística*. McGraw-Hill.

Kvanli (1988). *Statistics*. West Publishing Company.

Montgomery e Runger (2002). *Applied Statistics and Probability for Engineers*. Wiley.

Mood, Graybill e Boes (1974). *Introduction to the Theory of Statistics*. McGraw-Hill.

Natário (2010). *Notas de apoio à disciplina de Probabilidade e Estatística D*. DMAT.

Paulino e Branco (2005). *Exercícios de Probabilidade e Estatística*. Escolar Editora.

Rohatgi (1976). *An Introduction to Probability Theory and Mathematical Statistics*. Wiley.

Sokal e Rohlf (1995). *Biometry*. Freeman.

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

### Teaching method

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

### Evaluation method

Frequency: Obtained with at least two thirds of attendance in classes taught in each module.

CONTINUOUS EVALUATION

The students obtain approval if the weighted average of the three 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 = 40%T1 + 40%T2 + 20%T3

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