Probability and Statistics II


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





Responsible teacher

Filipe José Gonçalves Pereira Marques


Weekly - 5

Total - 70

Teaching language



Good knowledge of the materials taught in the course Probability and Statistics I


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

1. Pre-Requisites

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 0-20 scale) will be approved in the course.
  • Students who obtained a final grade from tests less than 9.5 (on a 0-20 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 0-20 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 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


Programs where the course is taught: