Course detail

# Probability, Statistics and 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.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

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.

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.

Planned learning activities and teaching methods

Techning methods include lectures and computer laboratories. Students have to write a four single homework during the course.

Assesment methods and criteria linked to learning outcomes

Students may be awarded
Up to 40 points for computer exercises (written test 20 points, 4 homework each max. 5 points).
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.
Requirements for successful completion of the are course provided in an annual public notice.

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. 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.

Work placements

Not applicable.

Aims

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

k dispozici na www.umat.feec.vutbr.cz/~fajmon/mpso
BAŠTINEC, J.; ZAPLETAL, J. Statistika, pravděpodobnost, operační výzkum. Statistika, pravděpodobnost, operační výzkum. Brno: 2007. s. 1-161.

Not applicable.

Classification of course in study plans

• Programme EEKR-M Master's

branch M-TIT , 1 year of study, winter semester, theoretical subject
branch M-BEI , 2 year of study, winter semester, theoretical subject
branch M-MEL , 1 year of study, winter semester, theoretical subject
branch M-EVM , 1 year of study, winter semester, theoretical subject
branch M-KAM , 1 year of study, winter semester, theoretical subject
branch M-EEN , 1 year of study, winter semester, theoretical subject
branch M-SVE , 1 year of study, winter semester, theoretical subject

• Programme EEKR-M Master's

branch M-SVE , 1 year of study, winter semester, theoretical subject
branch M-MEL , 1 year of study, winter semester, theoretical subject
branch M-EVM , 1 year of study, winter semester, theoretical subject
branch M-EEN , 1 year of study, winter semester, theoretical subject
branch M-TIT , 1 year of study, winter semester, theoretical subject
branch M-BEI , 2 year of study, winter semester, theoretical subject
branch M-KAM , 1 year of study, winter semester, theoretical subject

• Programme AUDIO-P Master's

branch P-AUD , 1 year of study, winter semester, elective interdisciplinary
branch P-AUD , 2 year of study, winter semester, elective interdisciplinary

• Programme EEKR-CZV lifelong learning

branch EE-FLE , 1 year of study, winter semester, theoretical subject

#### Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

1. Parameter estimation, t-test, confidence intervals.
2. Analysis of variance (ANOVA).
3. Correlation, regression.
4. After or instead of ANOVA.
5. Chi square distribution.
6. Nonparametric tests.
7. Linear programming.
8. Duality in linear programming.
9. Transport problem.
10.Dynamic programming.
11.Inventory models.
12.Probabilistic dynamic programming.

Exercise in computer lab

18 hod., compulsory

Teacher / Lecturer

Syllabus

In accordance with the lecture.