Graduate Program

__Credit hours__

In-class work per week |
Practice per week |
Credits |
Duration |
Total |

6 |
2 |
8 |
10 weeks |
120 hours |

__Instructor__

Adriano Júlio de Barros Vicente de Azevedo Filho

Ana Lucia Kassouf

__Objective__

The course is intended to provide basic concepts and theoretical foundations of probability and statistics

in order to prepare for disciplines and research in the areas of Econometrics, Economics of Uncertainty,

Game Theory, Decision and Risk Analysis, Simulation, Stochastic Processes and Probabilistic Modeling in

general. The main objectives of the course include training the student in understanding, modeling and

solving problems involving probabilities and statistical inference, aiming to improve their capacity for

analysis and abstraction in situations involving uncertainty.

__Content__

1. Deductive Inference x Inductive Inference: Basic concepts of formal logic, syllogisms, inductive

reasoning. Causality and logical implication. Alternative notions of statistics (classical and Bayesian

view). Understanding the meaning of axioms, theorems, lemmas, propositions, corollaries. Superficial

notions about the main methods used in proofs of theorems. 2. Algebra of Events and Probability

Axioms: Experiments, events, algebra of sets and events. Axioms of probability / conditional probability.

Bayes theorem. Discrete and continuous spaces. Probability trees. Notions of combinatorial analysis. 3.

Random Variables, Probability Distributions and Moments. Basics for discrete case and continuous case.

Density and distribution function (cumulative). General properties of distributions. Moment-generating

functions. Mathematical hope, variance, moments, measures of central tendency. Properties of hope and

variance. Multivariate distributions. Covariance. Conditional and marginal distributions. Conditional Hope

and Variance. Moment-generating functions and characteristic function. Regression. Independence.

Markov, Chebyshev and Jensen inequalities. 4. Modes of Convergence of Laws of Statistics: Law of Large

Numbers and Central Limit Theorem. 5. Parametric Probability Distributions. Notions and properties of

discrete (Bernoulli, Binomial, Poisson, Negative Binomial and others.) and continuous (Normal, Uniform,

Exponential, Log-Normal, Gamma, t-student, F, Chi-Square and others) distributions commonly used.

Approximations and mixtures of distributions. 6. Distributions of Random Variable Functions. Expectation

and variance. Approaches. Cumulative function technique, maximum and minimum distributions.

Technique of the moment generating function. Technique of transforming and changing variables.

Probability integral transformation theorem. Distributions of Functions of Random Variables by Monte

Carlo Simulation. Transformations in multivariate and Jacobian cases. 7. Estimates for Points.

Understanding estimators and statistics. Properties of estimators. Measures. Strategies for choosing

estimators. Least squares method. Method of moments. Bayesian Estimators. Estimation by maximum

likelihood: properties, Cramer-Rao limit, consistency, invariance, normality by convergence. 8.

Estimates for Intervals. Confidence intervals. Unilateral and bilateral intervals. Results for normal

samples, ranges for hope and variance. Bayesian ranges. 9. Testing and Hypothesis Selection. Basic

concepts. Types and dimensions of errors. Limitations. Decision theory and statistical tests Bayesian

notions. 10. Introduction to Econometric Regression Models. Basic principles. Least Squares Method.

Elementary Tests.

__Bibliography__

Amemiya, T. 1994. Introduction to Statistics and Econometrics. Harvard University Press, Cambridge.

Azevedo-Filho, A. 2010. Principios de Inferência Dedutiva e Indutiva: Noções de Lógica e Métodos de

Prova. CreateSpace (EUA). Azevedo-Filho, A. 2011. Introdução à Estatística Matemática Aplicada - Vol I:

Fundamentos. CreateSpace (EUA). Azevedo-Filho, A. 2011. Introdução à Estatistica Matemática Aplicada

- Vol II: Distribuições Paramétricas e Simulação. CreateSpace (EUA). Azevedo-Filho, A. 2007.

Probabilidades III - Distribuições Paramétricas Discretas e Contínuas. USP/DEAS - Série Didática, No. D-

133, 150p, Berger, J. 1993. Statistical Decision Theory and Decision Analysis, 2nd Edition, Springer,

617p, Berger, J. 2003. Could Fisher, Jeffreys and Neyman have agreed on testing? Statistical Science,

18:1-32, 2003 Berndt, E. 1996. The Practice of Econometrics: Classic and Contemporary. Addison-

Wesley, 702p. DeGroot, M. 1986. Probability and Statistics. Addison-Wesley. Drake, A. 1967.

Fundamentals of Applied Probabilistic Analysis. McGraw-Hill, New York. Greene, W. H. 2007.

Econometric Analysis. 6th Edition, Prentice Hall, Englewood Cliffs. Hoffmann, R. 2006. Estatística para

Economistas. 4ª. Edição, Editora Thomson Learning, São Paulo. Hoffmann, R. e Vieira, S. 1977. Análise

de Regressão: uma Introdução à Econometria. Hucitec-Edusp Hubbard, R. e Amstrong, J. 2006. Why we

don't really know what statistical significance means: Implications for educators. Journal of Marketing

Education, Volume 28, no. 2, 2006, Pages 114-120. Intriligator, Bodkin e Hsiao 1996. Econometric

Models, Techniques and Applications. 2nd Edition, Prentice-Hall. Mood, A., Graybill, F. A and Boes, D.

1989. Introduction to the Theory of Statistics. McGraw-Hill, New York. O'Haggan, A. e Luce, B. 2003. A

Primer on Bayesian Statistics. Centre for Bayesian Statistics in Health Economics, MEDTAP International,

2003. Spanos 1999. Statistical Foundations for Econometric Modelling with Observational Data.

Cambridge University Press.