Course detail

# Statistics, Stochastic Processes, Operations Research

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.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Entry knowledge

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.

Rules for evaluation and completion of the course

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 is optional.

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Baštinec, J.: Statistika, stochastické procesy, operační výzkum. 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.

Baštinec, J.: Statistika, stochastické procesy, operační výzkum. Sbírka příkladů. Brno 2017 (CS)
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
Sarma, R. D.:Basic Applied Mathematics for the Physical Sciences 3rd New edition Edition, 2017, ISBN-13: 978-8131787823

Classification of course in study plans

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, winter semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, winter semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, winter semester, elective

• Programme VTI-DR-4 Doctoral

branch DVI4 , any year of study, winter semester, elective

• Programme DPC-KAM Doctoral, any year of study, winter semester, compulsory-optional
• Programme DPC-EKT Doctoral, any year of study, winter semester, compulsory-optional
• Programme DPC-IBE Doctoral, any year of study, winter semester, compulsory-optional
• Programme DIT-EN Doctoral, any year of study, winter semester, compulsory-optional
• Programme DIT-EN Doctoral, any year of study, winter semester, compulsory-optional
• Programme DIT Doctoral, any year of study, winter semester, compulsory-optional
• Programme DIT Doctoral, any year of study, winter semester, compulsory-optional
• Programme DPC-MET Doctoral, any year of study, winter semester, compulsory-optional
• Programme DPC-SEE Doctoral, any year of study, winter semester, compulsory-optional
• Programme DPC-TLI Doctoral, any year of study, winter semester, compulsory-optional
• Programme DPC-TEE Doctoral, any year of study, winter semester, compulsory-optional

#### Type of course unit

Seminar

39 hours, optionally

Teacher / Lecturer

Syllabus

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.