Course details

# Statistics, Stochastic Procesess, Operational Analysis

DMA1 FEEC BUT DPC-MA1 Acad. year 2020/2021 Winter semester

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.

**State doctoral exam themes:**

- Random variable, distributional function, numerical characteristics of random variables.
- Distributions of random variables, the property normal distribution, law of three sigma.
- Limit theorems and by the reason of their uses in practice.
- Basic collection, random selection, its property and characteristic.
- Testing statistical hypotheses, the condition of usability of single tests.
- Graphics and algebraic technique of the solutions of the tasks of linear programming, the analysis of sensibility.
- By what differ not-linear programming from linear, technique at the solution?
- Markovské chain with discrete and continuous time and work with them.
- By the models of processes, Poisson model, linear model of the growth and of discontinuance.
- Evaluations of crossings, by the types deciding processes and of the way of their solutions.

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Subject specific learning outcomes and competences

The graduate of this course should be able to solve the optimizing and statistical problems of technical and economical practice.

Generic 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 kwnowledge 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.

Fundamental literature

- Zapletal, J.: Základy počtu pravděpodobnosti a matematrické statistiky. PC-DIR,VUT, Brno, 1995
- Anděl, J.: Statistické metody. Matfyzpress, MFF UK Praha, 1993.
- Taha, H.A.: Operations research. An Introduction. Fourth Edition, Macmillan Publishing Company, New York 1989.
- Miller, I., Miller, M.: John E. Freund's Mathematical Statistics. Sixth Edition. Prentice Hall, Inc., New Jersey 1999. Předchozí vydání publikováno pod názvem Freund, J.E.: Mathematical Statistics, Fifth Edition.
- Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers. Third Edition. John Wiley & Sons, Inc., New York 2003.
- Loftus, J., Loftus, E.: Essence of Statistics. Second Edition, A. A. Knopf, New York, 1988
- Baštinec, J.: Statistika, stochastické procesy, operační výzkum. Brno 2017
- Baštinec, J.: Statistika, stochastické procesy, operační výzkum. Sbírka příkladů. Brno 2017
- 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

Syllabus of lectures

- Classical and axiomatic definitions of probability. Conditional probability, total probability. Random variable, numerical characteristics.
- Discrete and continuous distributions of random variables. Properties of the normal distribution. Limit theorems.
- Statistics. Selection. Statistical processing of the material. Basic parameters and characteristics of the population selection.
- Basic point and interval estimates. Goodness. Analysis of variance.
- Operations Research. Linear programming. Graphic solution. Simplex method.
- Dual role. The sensitivity analysis. The economic interpretation of linear programming.
- Nonlinear programming.
- Solving of problems of nonlinear programming.
- Random processes, basic concepts, characteristics of random processes.
- Discrete Markov chain. Homogeneous Markov chains, classification of states. Regular Markov chains, limit vector, the fundamental matrix, and the median of the first transition.
- Absorption chain mean transit time, transit and residence. Analysis of Markov chains using Z-transform. Calculation of powers of the transition matrix.
- Continuous time Markov chains. Classification using the Laplace transform. Poisson process. Linear growth process, linear process of extinction, linear process of growth and decline.
- Markov decision processes. The award transitions. Asymptotic properties. Decision-making processes with alternatives. Hidden Markov processes.

Controlled instruction

Lectures are optional.

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 students ability to apply various methods of technical and economic practice.

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Course inclusion in study plans