Probability and Statistics II
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.
General characterization
Code
10978
Credits
9.0
Responsible teacher
Filipe José Gonçalves Pereira Marques
Hours
Weekly  5
Total  70
Teaching language
Português
Prerequisites
Good knowledge of the materials taught in the course Probability and Statistics I
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. McGrawHill, 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
1. PreRequisites
In order to be able to have access to the course evaluation, both to midterms and tests and also to the Exam, students need to have the presence in at least 2/3 of both Labs and Classes.
2. Evaluation
 The recommended form of evaluation consists in 3 Tests:
 1st Test  weight: 20%
 2nd Test  weight: 60%
 Project  weight: 20%
 The student who has an average grade (weighted mean) of at least 9,5 (on a 020 scale) will be approved in the course.
 Students who obtained a final grade from tests less than 9.5 (on a 020 scale), may have access to a final Exam, in case they have attended at least 2/3 of Labs and 2/3 of Classes.
 Also the students who had a grade equal or greater than 9.5 from tests may have access to the Final Exam in order to improve their grade.
 Students with a final grade of more than 17 (on a 020 scale) have to go through an oral examination, or their final grade will be equal to 17.
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 CramerRao 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. NeymanPearson Lemma
 Likelihood ratio tests.
Chap. 6 – Practical applications
 Confidence intervals and tests for the mean and variance of Normal populations.
 Chisquare tests of independence and goodnessoffit
 Goodnessoffit tests for the Normal distribution