Mathematics IV (FSI-4M-A)

Academic year 2023/2024
Supervisor: doc. RNDr. Libor Žák, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
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, encompassment of complete subject, classification sufficient or better of written exam and admission of semester assignment. Examination (written form) consists of two parts: a practical part (2 tasks from the theory of probability: probability and its properties, random variable, distribution Bi, H, Po, N and discrete random vector; 2 tasks from mathematical statistics: point and interval estimates of parameters, tests of hypotheses of distribution and parameters, linear regression model) using the summary of formula; a theoretical part (4 tasks related to basic notions, their properties, sense and practical use); evaluation: each task 0 to 20 points and each theoretical question 0 to 5 points; evaluation according to the total number of from examination and seminars: 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                  
    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, 2003.
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. Statistika a pravděpodobnost. Brno : FSI VUT v CERM, 2003.
2. Seger, J. - Hindls, R.: Statistické metody v tržním hospodářství. Praha : Victoria Publishing, 1995.
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-STI-A full-time study --- no specialisation -- Cr,Ex 5 Compulsory 1 2 S
B-STI-Z visiting student --- no specialisation -- Cr,Ex 5 Recommended course 1 1 S