Subject: Probability and Statistics

Scientific Area:

Mathematics

80 Hours

Number of ECTS:

7,5 ECTS

Language:

Portuguese

# Overall objectives:

1 - When successfully finishing the course the student: - will know the fundamental concepts, techniques and results of exploratory data analysis, probability and statistical inference; - will be able to read or consult studies using statistics; - will be able to present information and to develop simple statistical analysis, either in later courses of the program using the language of probability or statistical procedures, or in their professional activity.

# Syllabus:

1 - Introduction.
2 - Descriptive statistics and exploratory data analysis.
2.1 - Sample characteristics of location and scale.
2.2 - Grouping of data and graphical representations.
2.3 - Simple linear regression: estimation by the method of minimum squares and respective properties.
3 - Probability, conditional probability and independence.
3.1 - Frequentist and subjectivist probability concepts.
3.2 - Random experiments and events. Axiomatization of probability.
3.3 - Conditional probability and independence.
3.4 - Total probability theorem.
3.5 - Bayes theorem.
4 - Counts and measurements, discrete models and continuous models.
4.1 - Discrete random variables and probability mass function.
4.2 - Moments of discrete random variables.
4.3 - Discrete random pairs.
4.4 - Joint, marginal and conditional distributions; independence and covariance.
4.5 - Bernoulli variables.
4.6 - Binomial distribution.
4.7 - Geometric distribution.
4.8 - Negative binomial distribution.
4.9 - Hypergeometric distribution.
4.10 - The binomial distribution approximation to the hypergeometric distribution.
4.11 - Poisson distribution.
4.12 - The Poisson distribution approximation to the binomial distribution.
4.13 - Probability distribution function.
4.14 - Absolutely continuous random variables and probability density function.
4.15 - The exponential distribution.
4.16 - The uniform distribution.
4.17 - The probability integral transformation.
4.18 - Moments of absolutely continuous random variables.
4.19 - The normal distribution.
4.20 - The central limit theorem.
4.21 - The gamma function and its elementary properties.
4.22 - The gamma distribution.
4.23 - The moment-generating function.
4.24 - Sum of independent random variables.
4.25 - The chi-squared distribution.
5 - Introduction to statistical inference.
5.1 - Parameter estimation.
5.2 - Empirical sampling distributions of empirical moments of Gaussian populations.
5.3 - Estimation methods.
5.3.1 - The method of moments.
5.3.2 - The method of maximum likelihood.
5.4 - Confidence interval estimation in normal and in binomial populations.
5.5 - Hypothesis testing.
5.6 - Tests for means and variances in Gaussian populations and for proportions in binomial populations.
5.7 - The chi-squared goodness-of-fit test.

# Literature/Sources:

Wackerly D. D., Mendenhall W., Scheaffer R. L. , 2002 , Mathematical Statistics with Applications , Duxbury
Rohatgi, V. K. , 1976 , An Introduction to Probability Theory and Mathematical Statistics , John Wiley and Sons
Hogg, R. V., Craig, A. T. , 1995 , Introduction to Mathematical Statistics , Collier MacMillan
Ross, S. M. , 1989 , Introduction to Probability Models , Harcourt Academic Press
Spirer, H. J., Spirer, L., Jaffe, A. J. , 1998 , Misused Statistics , Marcel Dekker, Inc.
Pestana, D., Velosa, S. , 2010 , Introdução à Probabilidade e à Estatística, Vol. I, 4ª edição , Fundação Calouste Gulbenkian
Casella, G. and Berger, R. L. , 1990 , Statistical inference , Duxbury
Hoaglin, D. C., Mosteller, F., and Tukey, J. W. , 1992 , Análise Exploratória de Dados: Técnicas Robusta - Um Guia , Salamandra
Guttman, I., Wilks, S. S., and Hunter, J. H. , 1982 , Introductory Engineering Statistics , John Wiley & Sons
Montgomery, D. and Runger, G. , 1994 , Applied Statistics and Probability for Engineers , John Wiley & Sons
Feller, W. , 1976 , An Introduction to Probability Theory and Mathematical Statistics and Its Applications, Vol. I , John Wiley & Sons
Tufte, E. R. , 1997 , Visual explanations : images and quantities, evidence and narrative , Graphics Press
Tufte, E. R. , 2013 , The visual display of quantitative information , Graphics Press

# Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
This course has a theoretical and theoretical-practical component. The theoretical component aims to introduce the basic concepts, techniques and fundamental results of the discipline, properly exemplified, and will essentially be expository. In the theoretical-practical component students put into practice the knowledge acquired in the lectures by solving exercises that can be done individually or in small groups. The assessment model adopted is model A with two written tests, having equal weight in the final evaluation. Any of the tests can be recovered in the exam.