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 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.

Language of instruction

English

Number of ECTS credits

6

Mode of study

Not applicable.

Offered to foreign students

Of all faculties

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 BMA1 and BMA2 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 assessed by maximum 30 points (2 tests, maximum 15 points each). The exam can be taken only if at least the sum of 10 points has been obtained. Written exam is awarded by maximum 70 points.

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, 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., Mathematics 3. Electronic textbook. FEEC BUT. Brno. 2013 (EN)
M. Fusek, I. Hlavičková, M. Novák: Probability and Statistics: Exercise Book, Brno VUT, 2022 (EN)
M. Novák, Mathematics 3 (Numerical methods: Exercise Book), Brno, VUT, 2014 (EN)

Not applicable.

eLearning

Classification of course in study plans

• Programme BPA-ELE Bachelor's

specialization BPA-ECT , 2. year of study, winter semester, compulsory
specialization BPA-PSA , 2. year of study, winter semester, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Introduction to numerical methods. Numerical methods of solution of linear systems.
2. Numerical methods of solution of one non-linear equation and of non-linear systems.
3. Interpolation polynomials and splines.
4. Least squares method. Numerical differentiation and integration.
5. Introduction to probability theory.
6. Random variables and their numerical characteristics.
7. Random vectors and their numerical characteristics.
8. Selected probability distributions.
9. Law of large numbers, central limit theorem.
10. Introduction to statistics. Statistical processing of data.
11. Point and interval estimates. Method of moments and maximum likehood estimation.
12. Statistical tests with normal distribution, Pearson's chi-squared test.
13. Non-parametric tests.

Fundamentals seminar

4 hours, compulsory

Teacher / Lecturer

Exercise in computer lab

18 hours, compulsory

Teacher / Lecturer

Syllabus

1. Introduction to numerical methods. Numerical methods of solution of linear systems.
2. Numerical methods of solution of one non-linear equation and of non-linear systems.
3. Interpolation polynomials and splines.
4. Least squares method. Numerical differentiation and integration.
5. Introduction to probability theory.
6. Random variables and their numerical characteristics.
7. Random vectors and their numerical characteristics.
8. Selected probability distributions.
9. Law of large numbers, central limit theorem.
10. Introduction to statistics. Statistical processing of data.
11. Point and interval estimates. Method of moments and maximum likehood estimation.
12. Statistical tests with normal distribution, Pearson's chi-squared test.
13. Non-parametric tests.

Project

4 hours, compulsory

Teacher / Lecturer

eLearning