Course detail

# Mathematics 3

The aim of this course is to introduce the basics of two mathematical disciplines: numerical methods, and probability and statistics.
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.
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.

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 Mathematics 1 and Mathematics 2 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
- construct estimates of uknown parameters of the known distribution
- estimate parameters of a probability distribution by means of the maximum likelihood method.
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
- in all the above cases discuss principles of the respective methods, , choose a suitable method for a given task, discuss their convergence and justify one's reasoning

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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

Not applicable.

eLearning

Classification of course in study plans

• Programme BPC-AUD Bachelor's

specialization AUDB-ZVUK , 2. year of study, winter semester, compulsory
specialization AUDB-TECH , 2. year of study, winter semester, compulsory

• Programme BPC-EKT Bachelor's, 2. year of study, winter semester, compulsory
• Programme BPC-MET Bachelor's, 2. year of study, winter semester, compulsory
• Programme BPC-TLI Bachelor's, 2. year of study, winter semester, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Introduction to probability. Some probability models (classical, discrete, geometrical), conditional probability, dependence and independence of random events.
2. Random variables, random vector, distribution function.
3. Characteristics of random variables, basic distributions.
4. Characteristics of random vectors, covariance, correlation.
5. Law of large numbers, Central limit theorem.
6. Introduction to statistics, histogram,
7. Moment method, maximum likelihood method.
8. Numerical solution of systems of nonlinear equations. Systems of linear equations (Gaussian elimination with pivoting, Jacobi and Gauss-Seidel iterative methods).
9. Interpolation: interpolation polynomial (Lagrange and Newton), splines (linear and cubic)
10. Least squares approximation. Numerical differentiation.
11. Numerical integration (trapezoidal and Simpson method).
12. Numerical solution of differential equations: initial problems (Euler method and its modifications, Runge-Kutta methods), boundary value problems (very briefly).

Fundamentals seminar

4 hours, compulsory

Teacher / Lecturer

Computer-assisted exercise

18 hours, 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 hours, compulsory

Teacher / Lecturer

eLearning