Graduate Program

__Credit hours__

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

3 |
1 |
8 |
15 weeks |
120 hours |

__Instructor__

Edwin Moises Marcos Ortega

Fábio Prataviera

Idemauro Antonio Rodrigues de Lara

Renata Alcarde Sermarini

__Objective__

Train students in topics on Probability Theory and Statistical Inference.

__Content__

SUMMARY: Study and development of probabilistic and inference methods.

PROGRAM CONTENT: 1. Algebra and sigma-algebra of random events. 2. Basic principles of probability

theory: axioms and theorems. 3. Random variable, distribution function, probability function and

probability density function.4. Families of probabilistic models. 5. Random vectors, joint cumulative

distribution function, marginal and conditional functions, independence of random variables. 6.

Mathematical expectation, variance, covariance, conditional expectation, expectation of random vector

functions, moment generating function. 7. Transformation of random vectors: distribution function,

Jacobian and moment generating function techniques. 8. Basic concepts of inference, random sampling

and properties, order statistics and sampling distributions. 9. Principles of data reduction, sufficient and

complete statistics, k-parametric exponential family. 10. Point estimation methods: least squares,

moments and maximum likelihood. 11. Properties and comparison of estimators, unbiased estimator of

uniformly minimum variance, mean squared error, Rao-Blackwell and Lehmann Scheffé theorems. 12.

Interval estimation: confidence intervals for parameters associated with normal random variables,

pivotal quantity method. 13. Introduction to hypothesis testing: basic concepts, types of errors and

power function, uniformly more powerful tests.

__Bibliography__

1. Casella G.; Berger L.R. Statistical Inference. 2nd. Edition, Duxbury Press, 2002.

2. Degroot, M.H. Probability and Statistics. 4th. Edtion, Addison-Wesley, 2011.

3. James, B.R. Probabilidade: um curso em nível Intermediário. Projeto Euclides, RJ, 2008.

4. Magalhães, M.N. Probabilidade e Variáveis Aleatórias. 3ª Edição São Paulo, IME/USP, 2015.

5. Murteira, B.J.F. Probabilidade e Estatística. vol I, II. 2ª Edição, McGraw-Hill, Portugal, 1990.

6. Mood, A.M.; Graybill, F.A. and Boes, D. Introduction to the Theory of Statistics. 3rd. Edition, McGraw-

Hill, 1974.

7. Roussas, G.G. A Course in Mathematical Statistics. 2nd. Edition: Academic Press, 1997.

8. Sheldon, R. Introduction to Probability Models. Academic Press, 2014.

9. Zwanzig, S. Introduction to the Theory of Statistical Inference. Taylor & Francis, 2011.