Course detail

# Statistics, Stochastic Processes, Operations Research

FEKT-DKC-MA1Acad. year: 2021/2022

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.

Learning outcomes of the course unit

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.

Prerequisites

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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Baštinec, J.: Statistika, stochastické procesy, operační výzkum. Brno 2017 (CS)
Baštinec, J.: Statistika, stochastické procesy, operační výzkum. Sbírka příkladů. Brno 2017 (CS)
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.
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

Planned learning activities and teaching methods

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

Assesment methods and criteria linked to learning outcomes

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.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

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.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

Teaching is optional.

Classification of course in study plans

• Programme DKC-KAM Doctoral, any year of study, winter semester, 4 credits, compulsory-optional
• Programme DKC-EKT Doctoral, any year of study, winter semester, 4 credits, compulsory-optional
• Programme DKC-IBE Doctoral, any year of study, winter semester, 4 credits, compulsory-optional
• Programme DKC-MET Doctoral, any year of study, winter semester, 4 credits, compulsory-optional
• Programme DKC-SEE Doctoral, any year of study, winter semester, 4 credits, compulsory-optional
• Programme DKC-TLI Doctoral, any year of study, winter semester, 4 credits, compulsory-optional
• Programme DKC-TEE Doctoral, any year of study, winter semester, 4 credits, compulsory-optional

#### Type of course unit

Guided consultation

39 hours, optionally

Teacher / Lecturer