Probability and Statistics II
Code
10978
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
9.0
Teacher in charge
Carlos Manuel Agra Coelho
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
To give the students a solid knowledge in the areas of point and interval estimation, which may allow them to carry out and define testing procedures and confidence intervals for distribution and population parameters.
Prerequisites
Good knowledge of the materials taught in the course Probability and Statistics I
Subject matter
Chap. 1 - Convergence of random variables
- Convergence in distribution and in probability
- The weak law of large numbers
- Establishing convergence in distribution
- Central Limit Theorems.
- Convergence in distribution and convergence of moments
- The Continuity Theorem
- Convergence in mean of order h
- Convergence with probability 1
- The Strong Law of large numbers
Chap. 2 - On the distribution of some Sample Statistics
Chap. 3 – Parametric point estimation
-
Methods of estimation: Method of moments, Maximum Likelihood, Least Squares, Other
- Some desirable properties of estimators: Unbiasedness, Consistency, Invariance, Sufficiency, Completeness, Efficiency
- Unbiased estimation: BLUEs, UMVUEs, The Cramer-Rao lower bound
Chap. 4 – Parametric Interval Estimation
- Confidence intervals. Definition and examples.
- Confidence intervals for large samples
Chap. 5 – Tests of Hypotheses
- Some fundamental notions about tests of hypotheses.
- Most powerful test. Neyman-Pearson Lemma
- Likelihood ratio tests.
Chap. 6 – Practical applications
- Confidence intervals and tests for the mean and variance of Normal populations.
- Chi-square tests of independence and goodness-of-fit
- Goodness-of-fit tests for the Normal distribution
Bibliography
Coelho, C. A. . Tópicos em Probabilidades e Estatística, Vol. II, Chap. 8 e 9
Coelho, C. A. . Tópicos em Probabilidades e Estatística, Vol. III
Mood, A. M., Graybill, F. A. e Boes, D. C. (1974). Introduction to the Theory of Statistics, 3ªed. McGraw-Hill, New York
Rohatgi, V. K. (1976). An Introduction to Probability Theory and Mathematical Statistics, J. Wiley & Sons, New York
Teaching method
Lectures, where the main concepts and results will be introduced to the students, together with the presentation of illustrative examples, which are intended to enlighten the concepts presented. This Lectures will be complemented with Labs which will go shoulder to shoulder with the Lectures and where the resolution of problems, related to the concepts introduced in the Lectures, will be made with the active participation of the students.
Evaluation method
It will be necessary to have an attendance of at least 2/3 of both Classes (Theory) and Labs to be able to have access both the Tests and/or the Exam.
For those who will choose the evaluation through tests, there will be 3 tests:
- October 13, 2014 (in class) - weight=25% (duration: 1 hour)
- November 15, 2014 (out of class) - weight=40% (duration: 2 hours)
- December 17, 2014 (out of class) - weight=35% (and minimum grade of 7, on a 0-20 scale) (duration: 2 hours)
being the final grade (on a 0-20 scale) the weighted average of the classifications in the 3 tests (with the requirement for a minimal grade of 7 on the 3rd test). Approval is obtained with a grade of at least 10 (on a 0-20 scale).
For those who choose the evaluation by Exam or did not obtain a final grade of at least 10 in the evauation through tests:
- Final Exam