Probability and Statistics II (FSI-SP2)

Academic year 2018/2019
Supervisor: doc. RNDr. Libor Žák, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The course objective is to make students majoring in Mathematical Engineering acquainted with theoretical background of regression analysis and with real applications of regression methods in technical practice.
Learning outcomes and competences:
Students acquire needed knowledge from important parts of the probability theory and mathematical statistics, which will enable them to evaluate and develop stochastic models of technical phenomena and processes based on these methods and realize them on PC.
Prerequisites:
Rudiments of descriptive statistics, probability theory and mathematical statistics.
Course contents:
This course is concerned with the following topics: multidimensional normal distribution, linear regression model (estimates, tests of hypotheses, regression diagnostics), nonlinear regression model, introduction to ANOVA, categorial analysis, selected multivariate methods (correlation analysis). Students learn of the applicability of those methods and available software for computations.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
Course-unit credit requirements: active participation in seminars, mastering the subject matter, passing both written exams and semester assignment acceptance. Preparing and defending a project. Examination (semester assignment (12 points) and written form of exam (88 points) consisting of two parts: a practical part (4 tasks related to: random vectors, conditional distribution, multivariate normal distribution, regression analysis, categorial data analysis); theoretical part (4 tasks related to basic notions, their properties, sense and practical use, and proofs of two theorems); evaluation: each task 0 to 18 points and each theoretical question 0 to 4 points; evaluation according to the total number of points (scoring 0 points for any theoretical part task means failing the exam): excellent (90 - 100 points), very good (80 - 89 points), good (70 - 79 points), satisfactory (60 - 69 points), sufficient (50 - 59 points), failed (0 - 49 points).
Controlled participation in lessons:
Attendance at seminars is controlled and the teacher decides on the compensation for absences.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Computer-assisted exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture Random vector, moment characteristics.
Conditional distribution.
Characteristic function.
Multidimensional normal distribution - properties.
Distribution of quadratic forms.
Linear regression model (LRM) and parameter estimates in LRM
Testing hypotheses concerning linear regression model
Special case of LRM (regression line, regression parabola, polynomial regression, ANOVA models)
Weighted regression, an introduction into regression diagnostic and linearized regression model.
Goodness of fit tests with known and unknown parameters
Introduction to analysis of categorial data (contingency, chi-square test, measures of association, Fisher test).
Correlation analysis
    Computer-assisted exercise Random vector, variance-covariance matrix, correlation matrix.
Conditional distribution, conditional expectation, conditional variance.
Characteristic function - examples, properties.
Properties of multivariate normal distribution, linear transform.
Distributions of quadratic forms - examples for normal distributions.
Point and interval estimates of coefficients, variance and values of linear regression function.
Statistical software on PC
Testing hypotheses concerning linear regression functions: particular and simultaneous tests of coefficients, tests of model.
Multidimensional linear and nonlinear regression functions and diagnostics on PC.
Correlation coefficients, partial and multiple correlations.
Goodness of fit tests on PC.
Analysis of categorial data: contingency table, chi-square test, Fisher test.
Literature - fundamental:
1. Anděl, J.: Matematická statistika. Praha : SNTL, 1978.
2. Montgomery, D. C. - Runger, G.: Applied Statistics and Probability for Engineers, John Wiley & Sons, New York. 2002.
3. Lamoš, F. - Potocký, R.: Pravdepodobnosť a matematická štatistika. Bratislava : Alfa, 1989.
Literature - recommended:
1. Karpíšek, Z.: Matematika IV. Statistika a pravděpodobnost. Brno : FSI VUT v CERM, 2014.
2. Anděl, J.: Statistické metody. Praha : Matfyzpress, 2007.
3. Hebák, P. et al.: Vícerozměrné statistické metody (1), (2). Praha : Informatorium, 2004, 2005.
4. Zvára, K.: Regrese. Praha: Matfyzpress. 2008.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3A-P full-time study B-MAI Mathematical Engineering -- Cr,Ex 5 Compulsory 1 3 S