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

Czech, English

Completion

Credit+Examination (written)

Time span

26 hrs lectures, 13 hrs exercises, 13 hrs pc labs

Assessment points

70 exam, 10 half-term test, 20 labs

Department

Lecturer

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.

Prerequisites

Study literature

• Anděl, J.: Statistical Methods. Matfyzpress UK Praha, 1993.
• Diblík, J., Haluzíková, A., Baštinec, J.: Numerical Mathematics and Mathematical Statistics. SNTL Praha, 1987 (university text).
• Horová, I.: Numerical Methods. MU Brno, 1999.
• Nekvinda, M., Šrubař, J., Vild, J.: Introduction to Numerical Mathematics. SNTL Praha 1976.
• Vitásek, E.: Numerical Methods. SNTL Praha, 1987.
• Zapletal, J.: Grounding of Probability Calculus and Mathematical Statistics. PC-DIR Brno, 1995.

Fundamental literature

• Anděl, J.: Statistical methods. Matfyzpress UK Praha, 1993.
• Diblík, J., Haluzíková, A., Baštinec, J.: Numerical Mathematics and Mathematical Statistics. SNTL Praha, 1987 (university text).
• Horová, I.: Numerical Methods. MU Brno, 1999.
• Likeš, J., Machek, J.: Probability Calculus. SNTL Praha, 1987.
• Nekvinda, M., Šrubař, J., Vild, J.: Introduction to Numerical Mathematics. SNTL Praha 1976.
• Ralston, A.: A First Course in Numerical Analysis. Academia Praha, 1978.
• Vitásek, E.: Numerical Methods. SNTL Praha, 1987.
• Zapletal ,J.: Grounding of Probability Calculus and Mathematical Statistics. PC-DIR Brno, 1995.

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 - others, projects and individual work of students

• Topic will be given in the beginning of the term. Project must contain theory, solution in the form of a program and conclusion. Valuation: 20 points as a maximum.

Progress assessment

Written test, active work with MATLAB on computer practice.

Controlled instruction

Final exam ... 70 points.
Test ... 10 points.
Problems in MATLAB ... 20 points.

Exam prerequisites

Duty credit of the test, active work with MATLAB incl. prescribed results.