Course details
Probability and Numerical Methods
INM Acad. year 2004/2005 Winter semester 5 credits
Numerical mathematics: Metric spaces, Banach theorem. Solution of nonlinear equations. Approximations of functions, interpolation, least squares method, splines. Numerical derivative and integral. Solution of ordinary differential equations, one-step and multi-step methods. Probability: Random event and operations with events, definition of probability, independent events, total probability. Random variable, characteristics of a random variable. Probability distributions used, law of large numbers, limit theorems.
Guarantor
Language of instruction
Completion
Time span
- 26 hrs lectures
- 13 hrs exercises
- 13 hrs pc labs
Department
Subject specific learning outcomes and competences
Students apply the gained knowledge in technical courses when solving projects and writing the BSc thesis. Numerical methods represent the fundamental element of investigation and practice in the present state of research.
Learning objectives
In the first part the student will be acquainted with some numerical methods (approximation of functions, solution of nonlinear equations, approximate determination of a derivative and an integral, solution of differential equations) which are suitable for modelling various problems of practice. The other part of the subject yields fundamental knowledge from the probability theory (random event, probability, characteristics of random variables, probability distributions) which is necessary for simulation of random processes.
Prerequisite knowledge and skills
There are no prerequisites
Study literature
- Fajmon, B., Hlavičková, I., Novák, M., Vítovec, J.: Numerická matematika a pravděpodobnost (Informační technologie), VUT v Brně, 2014
- Hlavičková, I., Hliněná, D.: Matematika 3. Sbírka úloh z pravděpodobnosti. VUT v Brně, 2015
- Hlavičková, I., Novák, M.: Matematika 3 (zkrácená celoobrazovková verze učebního textu). VUT v Brně , 2014
- Novák, M.: Matematika 3 (komentovaná zkoušková zadání pro kombinovanou formu studia). VUT v Brně, 2014
- Novák, M.: Mathematics 3 (Numerical methods: Exercise Book), 2014
Fundamental literature
- Ralston, A.: Základy numerické matematiky. Praha, Academia, 1978.
- Horová, I.: Numerické metody. Skriptum PřF MU Brno, 1999.
- Maroš, B., Marošová, M.: Základy numerické matematiky. Skriptum FSI VUT Brno, 1997.
- Loftus, J., Loftus, E.: Essence of Statistics. Second Edition, Alfred A. Knopf, New York 1988.
- Taha, H.A.: Operations Research. An Introduction. Fourth Edition, Macmillan Publishing Company, New York 1989.
- Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers. Third Edition. John Wiley & Sons, Inc., New York 2003.
Syllabus of lectures
- Principle of numerical methods, error classification, accuracy improvement.
- Metric space, completeness, contraction, Banach fixed- point theorem.
- Solving of nonlinear equations.
- Approximation, interpolation polynomial, least squares method, spline.
- Numerical derivative and integral, composite quadrature formulae.
- Solving of ordinary differential equations, one-step methods.
- Multi-step methods.
- Elementary event, operation with events, field of events.
- Definition of probability, conditional probability, event independence, total probability theorem.
- Random variable, distribution function, random variable distribution, probability density.
- Two-dimensional random variable, random variable characteristic.
- Some important distributions, law of large numbers, limit theorems.
- Fundamental concepts, hypothesis testing.
Syllabus of numerical exercises
- Numerical error estimates, Richardson extrapolation.
- Interpolation polynomial.
- Application of Banach theorem.
- Probability.
- Distribution function, probability density.
- Normal distribution.
- Numerical characteristics.
Syllabus of computer exercises
- Solving of nonlinear equations.
- Approximation of functions.
- Spline.
- Numerical integration.
- Solving of differential equations.
- Fundamental types of probability distribution.
Progress assessment
Duty credit of the test, active work with MATLAB incl. prescribed results.
Controlled instruction
Final exam ... 70 points.
Test ... 10 points.
Problems in MATLAB ... 20 points.