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

Frederico Almeida Gião Gonçalves Caeiro


Weekly - 4

Total - 56

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

Classes are theoretical/practical with oral presentation of concepts, methodologies, and examples, complemented with problem solving. Specific student difficulties will be addressed during classes or in individual sessions scheduled with the professor.
Students need to attend a minimum of two thirds of the classes in order to be evaluated. Continuous evaluation is based on two or three tests. If a student does not obtain approval through continuous evaluation he can try it in an additional assessment.

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 on the practical and presential classes.

  2. Evaluation 

  • The recommended form of evaluation consists in:
    • 1st Test - weight: 50%
    • 2nd Test - weight: 50%
    • 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