Probability, Statistics and Operations Research
FEKT-LPSOAcad. year: 2019/2020
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.
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.
• Describe the work with statistical tables.
• 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.
• Calculate the optimal solution transport tasks and task assignment optimal solution.
• List the different models in stocks reserve.
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.
Recommended optional programme components
Recommended or required reading
J.Loftus, E.Loftus: Essence of Statistics. Second Edition. Alfred A.Knopf, New York 1988.
H.A.Taha: Operations research. An introduction. Fourth Edition. Macmilan Publishing Company, New York 1989.
J.Zapletal: základy počtu pravděpodobnosti a matematické statistiky. Skriptum FEI VUT, PC DIR Brno 1995.
B.Fajmon: Přednáškový text předmětu SMPO - oskenovaný rukopis přednášek. 213 stran, na adrese wes.feec.vutbr.cz/~fajmon/mpso/mpso.zip.
Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers. Third Edition. John Wiley \& Sons, Inc., New York 2003.
BAŠTINEC, J.; ZAPLETAL, J. Statistika, pravděpodobnost, operační výzkum. Statistika, pravděpodobnost, operační výzkum. Brno: 2007. s. 1-161.
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 40 points for work during semester, i.e. 4 homework tasks (maximum 5 points each) and up to 20 points for one practical computer homework tasks.
Up to 60 points for the written final exam. The test contains both theoretical and numerical tasks that are used to verify the orientation of students in statistics and operations research. Numerical tasks are included to verify the student's ability to apply statistical and optimization methods in technical and economic practice.
Language of instruction
The subject consists of 5 tutorials and 2 practical computer classes. The distribution of topics respects topics of MPSO. The last tutorial is devoted to repetition and preparations for the exam. In computer classes examples and possible ways of programming the topics in MATLAB and Excel are included.
MPSO topics structure:
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. t-test, F-test. The confidence intervals.
5. Linear regression. Post-hoc tests. Goodness.
6. Analysis of variance.
7. Paired test, unpaired test.
8. Non-parametric tests.
9. Operations Research. Linear programming. Graphic solution. Simplex method.
10. Dual role. The sensitivity analysis.
11. The economic interpretation of linear programming.
12. Transport role. Assignment task.
13. Dynamic programming, recursive algorithms, models in stocks reserve.
The objecive of the course is to enlarge the knowledge in the area of statistical tests and confidence intervals, to show some spheres of mathematical thinking in economics and to introduce the concepts of recursive algorithms.
Specification of controlled education, way of implementation and compensation for absences
Computer exercises are compulsory. Properly excused absence can be replaced by individual homework, which focuses on the issues discussed during the missed exercise.
Specifications of the controlled activities and ways of implementation are provided in annual public notice.
Date of the written test is announced in agreement with the students at least one week in advance. The new term for properly excused students is usually during the credit week.