Course detail

Mathematics 3

FEKT-BPC-MA3AAcad. year: 2023/2024

The aim of this course is to introduce the basics of two mathematical disciplines: numerical methods, and probability and statistics.
In the field of probability, main attention is paid to random variables, both discrete and continuous. The end of the course of probability is devoted to hypothesis testing.
In the field of numerical mathematics, the following topics are covered: root finding, systems of linear equations, curve fitting (interpolation and splines, least squares method), numerical differentiation and integration, numerical solving of differential equations.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Entry knowledge

The student should be able to apply the basic knowledge of combinatorics on the secondary school level: to explain the notions of variations, permutations and combinations, to determine their counts, to perform computations with factorials and binomial coefficients.
From the AMA1 and AMA2 courses, the basic knowledge of differential and integral calculus is demanded. Especially, the student should be able to sketch the graphs of elementary functions, to substitute into functions, to compute derivatives (including partial derivatives) and integrals.

Rules for evaluation and completion of the course

Students' work during the semestr is awarded by maximum 30 points (20 points for 1 test and 10 points for 1 project). The exam can be taken only if the score is 10 or better. Written exam is awarded by maximum 70 points. In case of coronavirus restrictions resulting in the impossibility to write the test in person at faculty premises, work during the semester will be awarded by maximum 10 points (1 project) and the exam will be awarded by maximum 90 points.
The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Aims

The aim of this course is to introduce the basics of two mathematical disciplines: numerical methods, and probability and statistics.
Students completing this course should be able to:
In the field of probability and statistics:
- compute the basic characteristics of statistical data (mean, median, modus, variance, standard deviation)
- choose the correct probability model (classical, discrete, geometrical probability) for a given problem and compute the probability of a given event
- compute the conditional probability of a random event A given an event B
- recognize and use the independence of random events when computing probabilities
- apply the total probability rule and the Bayes' theorem
- work with the cumulative distribution function, the probability mass function of a discrete random variable and the probability density function of a continuous random variable
- construct the probability mass functions (in simple cases)
- choose the appropriate type of probability distribution in model cases (binomial, hypergeometric, exponential, etc.) and work with this distribution
- compute mean, variance and standard deviation of a random variable and explain the meaning of these characteristics
- perform computations with a normally distributed random variable X: find probability that X is in a given range or find the quantile/s for a given probability
- approximate the binomial distribution with help of the normal distribution
- perform simple hypothesis testing: Z-test, test on the mean of normal distribution variance known, test on the parameter p of the binomial distribution

In the field of numerical methods, the student should be able to:
- find the root of a given equation f(x)=0 using the bisection method, Newton method or the iterative method, describe these methods including the convergence conditions
- find the root of a system of two equations using Newton or iterative method
- solve a system of linear equations using Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iteration methods, discuss the advantages and disadvantages of these methods
- find Lagrange or Newton interpolation polynomial for given points and use it for approximating the given function
- find the approximation of a function by spline functions
- find the approximation of a function given by table of points by the least squares method (linear, quadratic or exponential approximation)
- choose the most convenient type of approximation (interpolation polynomial, spline, least squares)
- estimate the derivative of a given function using numerical differentiation
- compute the numerical approximation of a definite integral using trapezoidal and Simpson method, describe the principal of these methods, compare them according to their accuracy
- find the approximate solution of a differential equation using Euler method, modified Euler methods and Runge-Kutta methods

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Fajmon, B., Hlavičková, I., Novák, M. Matematika 3. Elektronický text FEKT VUT, Brno, 2013 (CS)

Recommended reading

Hlavičková, I., Hliněná, D. Matematika 3 - Sbírka úloh z pravděpodobnosti. Elektronický text FEKT VUT, Brno, 2015 (CS)
Novák, M., Matematika 3 - Sbírka příkladů z numerických metod. Elektronický text FEKT VUT, Brno, 2015 (CS)

Elearning

Classification of course in study plans

  • Programme BPC-BTB Bachelor's 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

1. Introduction to numerical mathematics. Linear systems.
2. Numerical solution of nonlinear equations.
3. Nonlinear systems.
4. Interpolation polynomial. Spline.
5. Least squares method. Numerical differentiation and integration.
6. Numerical solutions of initial value problems.
7. Introduction to probability theorz.
8. Random variable and its numerical characteristics.
9. Significant discrete distributions.
10. Significant continuous distributions.
11. Normal distribution. Central limit theorem.
12. Introduction of statistical testing.
13. Revision, consultations.

Fundamentals seminar

4 hod., compulsory

Teacher / Lecturer

Syllabus

V hodinách je procvičována látka probraná na přednášce. (s přihlédnutím ke státním svátkům)

Computer-assisted exercise

18 hod., compulsory

Teacher / Lecturer

Syllabus

1. Combinatorics, classical and geometrical probability
2. Conditional probability, total probability rule and Bayes theorem
3. Discrete random variables, discrete distributions
4. Continuous random variables
5. Normal distribution, normal approximation to binomial distribution
6. Hypothesis testing
7. Root separation, bisection, Newton and iterative methods
8. Interpolation polynomial, spline functions
9. Least squares method
10. Numerical differentiation and integration
11. Numerical solution of differential equations - Euler and Runge-Kutta methods

Project

4 hod., compulsory

Teacher / Lecturer

Syllabus

Zpracování projektu (průřez tématy z numerických metod a pravděpodobnosti a statistiky).

Elearning