Course details

# Statistics, Stochastic Processes, Operations Research

DPC-MA1 FEKT DPC-MA1 Acad. year 2022/2023 Winter semester 4 credits

The course focuses on consolidating and expanding students' knowledge of probability theory, mathematical statistics and theory of selected methods of operations research. Thus it begins with a thorough and correct introduction of probability and its basic properties. Then we define a random variable, its numerical characteristics and distribution. On this basis we then build descriptive statistics and statistical hypothesis testing problem, the choice of the appropriate test and explanation of conclusions and findings of tests. In operational research we discuss linear programming and its geometric and algebraic solutions, transportation and assignment problem, and an overview of the dynamic and probabilistic programming methods and inventories. In this section the illustrative examples are taken primarily from economics. In the next the course includes an introduction to the theory of stochastic processes types. Therefore, it starts with repetition of necessary mathematical tools (matrices, determinants, solving equations, decomposition into partial fractions, probability). Then we construct the theory of stochastic processes, where we discuss Markovský processes and chains, both discrete and continuous. We include a basic classification of state and students learn to determine them. Great attention is paid to their asymptotic properties. The next section introduces the award transitions between states and students learn the decision-making processes and their possible solutions. In conclusion, we mention the hidden Markov processes and possible solutions.

Guarantor

Language of instruction

Completion

Time span

- 39 hrs seminar

Department

Instructor

Subject specific learning outcomes and competences

After completing the course the student will be able to:

• Describe the role of probability using set operations.

• Calculate basic parameters of random variables, both continuous and discrete ones.

• Define basic statistical data. List the basic statistical tests.

• Select the appropriate method for statistical processing of input data and perform statistical test.

• Explain the nature of linear programming.

• Convert a word problem into the canonical form and solve it using a suitable method.

• Perform sensitivity analysis in a geometric and algebraic way.

• Convert the specified role into its dual.

• Explain the difference between linear and nonlinear programming.

• Describe the basic properties of random processes.

• Explain the basic Markov property.

• Build an array of a Markov chain.

• Explain the procedure to calculate the square matrix.

• Perform the classification of states of Markov chains in discrete and continuous case.

• Analyze a Markov chain using the Z-transform in the discrete case and the Laplace transformation in the continuous case.

• Explain the procedure for solving decision problems.

• Describe the procedure for solving the decision-making role with alternatives.

• Discuss the differences between the Markov chain and hidden Markov chain.

Learning objectives

The aim of the subject is to deepen and broaden students' knowledge of statistical data processing and statistical tests. To provide students with basic knowledge in the field of operations research a teach them to use some optimization methods suitable for use in e.g. economics. Next is to provide students with a comprehensive overview of the basic concepts and results relating to the theory of stochastic processes and especially Markov chains and processes. We show possibilities of application of the decision-making processes of various types.

Prerequisite knowledge and skills

We require knowledge at the level of bachelor's degree, i.e. students must have proficiency in working with sets (intersection, union, complement), be able to work with matrices, handle the calculation of solving systems of linear algebraic equations using the elimination method and calculation of the matrix inverse, know graphs of elementary functions and methods of their design, differentiate and integrate of basic functions.

Study literature

- Miller, I., Miller, M.: John E. Freund's Mathematical Statistics. 8th Edition. Prentice Hall, Inc., New Jersey 2012.
- Taha, H.A.: Operations research. An Introduction. 9th Edition, Macmillan Publishing Company, New York 2013.ISBN-13: 978-0132555937
- Anděl, J.: Statistické úlohy, historky a paradoxy. Matfyzpress, MFF UK Praha, 2018.
- Zapletal, J.: Základy počtu pravděpodobnosti a matematrické statistiky. PC-DIR,VUT, Brno, 1995
- Papoulis, A., Pillai, S. U.: Probability, Random Variables and Stochastic Processes, 4th Edition, 2012. ISBN-13: 978-0071226615
- Nagy, I.: Základy bayesovského odhadování a řízení, ČVUT, Praha, 2003
- Sarma, R. D.:Basic Applied Mathematics for the Physical Sciences 3rd New edition Edition, 2017, ISBN-13: 978-8131787823

Fundamental literature

- Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for engineers. 6th Edition. John Wiley \& Sons, Inc., New York 2015.ISBN-13: 978-1118539712.

Syllabus of seminars

1. Classical and axiomatic definitions of probability. Conditional probability, total probability. Random variable, numerical characteristics.

2. Discrete and continuous distributions of random variables. Properties of the normal distribution. Limit theorems.

3. Statistics. Selection. Statistical processing of the material. Basic parameters and characteristics of the population selection.

4. Basic point and interval estimates. Goodness. Analysis of variance.

5. Operations Research. Linear programming. Graphic solution. Simplex method.

6. Dual role. The sensitivity analysis. The economic interpretation of linear programming.

7. Nonlinear programming.

8. Solving of problems of nonlinear programming.

9. Random processes, basic concepts, characteristics of random processes.

10. Discrete Markov chain. Homogeneous Markov chains, classification of states. Regular Markov chains, limit vector, the fundamental matrix, and the median of the first transition.

11. Absorption chain mean transit time, transit and residence. Analysis of Markov chains using Z-transform. Calculation of powers of the transition matrix.

12. Continuous time Markov chains. Classification using the Laplace transform. Poisson process. Linear growth process, linear process of extinction, linear process of growth and decline.

13. Markov decision processes. The award transitions. Asymptotic properties. Decision-making processes with alternatives. Hidden Markov processes.

Syllabus of seminars

I. Statistics (5 weeks)

Basic notions from probability and statistics. Statistical sets. Point and interval estimates.Testing hypotheses with parametres (not only for normal distribution). Tests of the form of distribution. Regression analysis. Tests of good accord. Non-parametric tests.

II. Stochastic processes(4 weeks)

Deterministic and stochastic problems. Characteristics of stochastic processes. Limit, continuity, derivation and integral of a stochastic process. Markov, stationary, and ergodic processes. Canonical and spectral division of a stochastic process.

III. Operation analysis (4 weeks)

Principles of operation analysis, linear and nonlinear programming. Dynamic programming, Bellman principle of optimality. Theory of resources. Floating averages and searching hidden periods.

Progress assessment

Students may be awarded

Up to 100 points for the final exam, which consists of writen and oral part. Entering the written part of the exam includes theoretical and numerical task that are used to verify the orientation of student in statistic, operation research and stochastic processes. Taking numerical task is to verify the student’s ability to apply various methods of technical and economic practice.

Teaching methods and criteria

Teaching methods depend on types of classes. They are described in Article 7 of the Study and Examination Regulations of Brno University of Technology.

Controlled instruction

Teaching is optional.

Course inclusion in study plans

- Programme DIT, any year of study, Compulsory-Elective group T
- Programme DIT, any year of study, Compulsory-Elective group T
- Programme DIT-EN (in English), any year of study, Compulsory-Elective group T
- Programme DIT-EN (in English), any year of study, Compulsory-Elective group T
- Programme VTI-DR-4, field DVI4, any year of study, Elective
- Programme VTI-DR-4, field DVI4, any year of study, Elective
- Programme VTI-DR-4 (in English), field DVI4, any year of study, Elective
- Programme VTI-DR-4 (in English), field DVI4, any year of study, Elective