Mathematics IV (FSI-4M)

Academic year 2023/2024
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 acquainted with basic notions, methods and progresses of probability theory, descriptive statistics and mathematical statistics as well as with the development of students` stochastic way of thinking for modelling a real phenomenon and processes in engineering branches.
Learning outcomes and competences:
Students obtain the needed knowledge of the probability theory, descriptive statistics and mathematical statistics, which will enable them to understand and apply stochastic models of technical phenomena based upon these methods.
Prerequisites:
Rudiments of the differential and integral calculus.
Course contents:
The course makes students familiar with descriptive statistics, random events, probability, random variables and vectors, probability distributions, random sample, parameters estimation, tests of hypotheses, and linear regression analysis. Seminars include solving problems and applications related to mechanical engineering. PC support is dealt with in the course entitled Statistical Software, which is optional.
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:

Seminar credit conditions:

  • active attendance in practices
  • sufficient or better of written exam
  • admission of semester assignment.

Examination (written form) consists of two parts:

  • a practical part – tasks from the covered topics. Total 0 to 80 points (with own summary of formulas)
  • a theoretical part - 4 tasks related to basic notions, (their properties, sense and practical use); each theoretical question 0 to 5 points.

Evaluation: each task 0 to 20 points evaluation according to the total number of points from practical and theoretical part: 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).

Attendance at seminars is controlled and the teacher decides on the compensation for absences.
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                  
    Exercise  13 × 2 hrs. compulsory                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 1. Random events, probability, conditional probability, independent events.
2. Random variable, functional characteristics.
3. Numerical characteristics of random variables. Introduction to basic distributions.
4. Basic distributions – continuation (properties and application).
5. Random vector, types, functional and numerical characteristics.
6. Random sample, sample characteristics (properties, sample from N).
7. Parameters estimation (point and interval estimates of parameters N and Bi).
8. Testing statistical hypotheses (types, basic concepts) – one-samle tests.
9. Testing statistical hypotheses – two-samle tests.
10. Testing statistical hypotheses – multi-samle tests, goodness of fit tests.
11. Elements of regression analysis. – introduction, point estimates
12. Regression analysis – interval estimates, hypotheses testing.
13. Regression analysis – diagnostics, other regression models, Summary of covered topics.
    Exercise 1. Descriptive statistics (one-dimensional sample).
2. Descriptive statistics (two-dimensional sample). Combinatorics. Semester work assignment.
3. Probability, conditional probability, independent events.
4. System reliability, random variables - functional and numerical characteristics.
5. Random variables – continuation.
6. Basic probability distributions (Minitab – reliability diagrams).
7. Two-dimensional discrete random vector, functional and numerical characteristics. CLT illustration
8. Written exam. Interval estimates.
9. Testing hypotheses of one-dimensional parameters, power of tests.
10. Testing hypotheses of parameters in two samples.
11. ANOVA, chisquare tests (equality of probabilities of more categories), goodnes of fit tests.
12. Regression analysis - regression line.
13. Regression analysis – model, assignment submition.
    Computer-assisted exercise Computer seminars follow up the topics covered in seminars using statistical software (Excel, Minitab).
Literature - fundamental:
1. Montgomery, D. C. - Renger, G.: Applied Statistics and Probability for Engineers. New York : John Wiley & Sons, 2017.
2. Hahn, G. J. - Shapiro, S. S.: Statistical Models in Engineering.New York : John Wiley & Sons, 1994.
3. Anděl, J.: Základy matematické statistiky. Praha : Matfyzpress, 2005.
Literature - recommended:
1. Karpíšek, Z.: Matematika IV. Pravděpodobnost a statistika. Učební text FSI VUT v Brně. Akademické nakladatelství CERM: Brno, 2003.
2. Karpíšek, Z., Drdla, M.: Applied Statistics. Textbook. Brno : FME BUT, 2007. File ApplStat2007.pdf .
3. Meloun, M. - Militký, J.: Statistické zpracování experimentálních dat. Praha : Plus, 1994.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-ZSI-P full-time study STI Fundamentals of Mechanical Engineering -- Cr,Ex 5 Compulsory 1 2 S
B-ZSI-P full-time study MTI Materials Engineering -- Cr,Ex 5 Compulsory 1 2 S
N-PMO-P full-time study --- no specialisation -- Cr,Ex 5 Compulsory-optional 2 1 S
B-FIN-P full-time study --- no specialisation -- Cr,Ex 5 Compulsory 1 2 S
B-MET-P full-time study --- no specialisation -- Cr,Ex 5 Compulsory 1 2 S