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Probability and Statistics B

Objectives

The aim of the course is to provide students with basics in Probability and Statistics allowing them to deal with uncertainty and variation, features that are part of every engineering project. In particular,

–  Build probabilistic models for (random) experiments;

–  Use the main probability distributions;

–  Explore and analyze collected data;

–  Estimate and testing population parameters based on sample data.

General characterization

Code

12510

Credits

6.0

Responsible teacher

João Filipe Lita da Silva

Hours

Weekly - 4

Total - 76

Teaching language

Português

Prerequisites

Basics in Real Analysis and Set Theory.

Bibliography

[1] A Course in Mathematical Statistics, G.G. Roussas, Academic Press, 1997.

 

[2] An Introduction to Probability and Statistics, V.K. Rohatgi, A.K.Md. Ehsanes Saleh, John Wiley & Sons, Inc., 2001.

 

[3] Applied Statistics and Probability for Engineers, D.C. Montgomery, G.C. Runger, John Wiley & Sons, Inc., 2011.

 

[4] Cours de Statistique Descriptive, G. Calot, Dunod, 1965.

 

[5] Introdução à Estatística, B. Murteira, C.S. Ribeiro, J. Andrade e Silva, C. Pimenta, McGraw-Hill, 2002.

 

[6] Introductory Statistics, P.S. Mann, John Wiley & Sons, Inc., 2010.

 

[7] Probabilités, Analyse des Données et Statistique, G. Saporta, Editions Technip, 2006.

 

[8] Statistics and Probability with Applications for Engineers and Scientists using Minitab, R, and JMP, B.C. Gupta, I. Guttman, K.P. Jayalath, John Wiley & Sons, Inc., 2020.

Teaching method

The theoretical results are presented to the students in formal lectures (with sketch of the proof of main statements), along with several enlightening and commented applications. In each lecture, proposed problems are explored and solved, involving the students on this process.

Evaluation method

1. The student''s evaluation is determined by one (and one only) component: theoretical-practical.

2. The theoretical-practical component encloses either two tests or a final exam.   

(a) Each test is graded in the interval [0,20] and it establishes 50% of the final grade.

(b) Alternatively, the final exam sets up 100% of the final grade and it is graded in the interval [0,20].

3. To pass, the student must get, rounded to units, a final grade greater or equal to 10.

Subject matter

1. Elements of probability

1.1 Counting techniques. Sets

1.2 Sample spaces. Events

1.3 Probability measure. Classical definition of probability and geometric probability

1.4 Conditional probability

1.5 Multiplication rule and Bayes’ theorem

1.6 Independence of events

 

2. Random variables and distributions functions

2.1 Random variables

2.2 Cumulative distribution functions

2.3 Functions of a random variable

2.4 Multiple random variables

2.5 Join cumulative distribution functions and marginal distributions

2.6 Independent random variables

 

3. Discrete and (absolutely) continuous random variables

3.1 Discrete random variables

3.2 Expectation, variance, and moments of discrete random variables

3.3 Some discrete random variables

3.4 (Absolutely) continuous random variables

3.5 Expectation, variance, and moments of (absolutely) continuous random variables

3.6 Some (absolutely) continuous random variables. Normal distribution (Gaussian law)

 

4. Two-dimensional discrete and (absolutely) continuous random variables 

4.1 Two-dimensional discrete random variables

3.2 Covariance, correlation, and moments of two-dimensional discrete random variables

3.3 Two-dimensional (absolutely) continuous random variables

3.4 Covariance, correlation, and moments of two-dimensional (absolutely) continuous random variables

3.5 Bivariate normal distribution

3.6 Orthogonal random variables. Inequalities 

 

5. Central limit theorem

5.1 Sums of independent random variables

5.2 Central limit theorem

 

6. Descriptive statistics

6.1 Qualitative and quantitative data

6.2 Samples. Population

6.3 Frequency distributions. Hystograms

6.4 Numerical measures: centrality, dispersion, and relative position

6.5 Box-whisker plot

6.6 Measures of association

 

7. Sampling distributions

7.1 Random sampling

7.2 (Sample) statistic

7.3 Sample mean and sample variance. Sampling distributions

 

8. Estimation

8.1 Point estimation

8.2 Point estimators. Properties

8.3 Moment estimators

8.4 Satistical intervals

8.5 Confidence intervals for the mean and variance

 

9. Hypothesis testing

9.1 Basic concepts. Statistical hypothesis, type I error, and type II error

9.2 Tests of statistical hypothesis. One-sided and two-sided hypothesis

9.3 p-value

9.4 Hypothesis tests on the mean. Hypothesis tests on the variance

9.5 Hypothesis tests on the difference of means

 

10. Analysis of variance

10.1 One-way ANOVA: fixed-effect model and assumptions

10.2 Two-way ANOVA: crossed classification without interaction