Course detail

# Mathematics

ÚSI-ESMATAcad. year: 2020/2021

Basic mathematical notions. Concept of a function, sequences, series. Vector spaces (linear combination of vectors, linear dependence, independence vectors, base, dimension of a vector space). Matrices and determinants. Systems of linear equations and their solution. Differential calculus of one variable, limit, continuity, derivative of a function. Derivatives of higher orders, l´Hospital rule, behavior of a function. Integral calculus of fuctions of one variable, indefinite integral. Integration by parts, substitution methods. Definite integral and its applications. Introduction to descriptive statistics. Introduction to probability. Some probability models (classical, discrete, geometrical), conditional probability, dependence and independence of random events. Total probability rule and Bayes theorem. Discrete random variables (probability mass function, cumulative distribution function, mean and variance). Discrete probability distributions (binomial, geometric, hypergeometric, Poisson, uniform). Continuous random variables (probability density function, distrubution function, mean, variance, quantiles). Exponencial distribution. Normal distribution. Central limit theorem. Testing of statistical hypotheses (t-test).

Learning outcomes of the course unit

After completing the course, students should be able to:
- decide whether vectors are linearly independent and whether they form a basis of a vector space;
- add and multiply matrices, compute the determinant of a square matrix, compute the rank and the inverse of a matrix;
- solve a system of linear equations;
- estimate the domains and sketch the grafs of elementary functions;
- compute limits and asymptots for the functions of one variable, use the L’Hospital rule to evaluate limits;
- differentiate and find the tangent to the graph of a function, find the Taylor ploynomial of a function near a given point;
- sketch the graph of a function including extrema, points of inflection and asymptotes;
- integrate using technics of integration, such as substitution and integration by parts;
- evaluate a definite integral including integration by parts and by a substitution for the definite integral;
- compute the area of a region using the definite integral;
- 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
- perform a simple hypothesis testing (t-test)

Prerequisites

Knowledge within the scope of standard secondary school requirements.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Krupková, V., Fuchs, P.,: Matematika 1. Elektronický text FEKT VUT, Brno, 2014
Fajmon, B., Hlavičková, I., Novák, M. Matematika 3. Elektronický text FEKT VUT, Brno, 2014
Kolářová, E: Matematika 1 - Sbírka úloh, 2010
Švarc, S. a kol., Matematická analýza I, PC DIR, Brno, 1997.
Brabec B., Hrůza,B., Matematická analýza II, SNTL, Praha, 1986.
Neubauer, J., Sedlačík, M., Kříž, O.: Základy statistiky. Praha: Grada Publishing, 2012.
Likeš, J., Machek, J.: Počet pravděpodobnosti. Praha: SNTL - Nakladatelství technické literatury, 1981.
Casella, G., Berger, R. L.: Statistical Inference. Pacific Grove, CA: Duxbury Press, 2001.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Written examination. Requirements for completion of the course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Basic mathematical notions. Concept of a function, sequences, series.
2. Vector spaces (linear combination of vectors, linear dependence, independence vectors, base, dimension of a vector space).
3. Matrices and determinants.
4. Systems of linear equations and their solution.
5. Differential calculus of one variable, limit, continuity, derivative of a function.
6. Derivatives of higher orders, l´Hospital rule, behavior of a function.
7. Integral calculus of fuctions of one variable, indefinite integral. Integration by parts, substitution methods.
8. Definite integral and its applications.
9. Introduction to descriptive statistics
10. Introduction to probability. Some probability models (classical, discrete, geometrical), conditional probability, dependence and independence of random events. Total probability rule and Bayes theorem.
11. Discrete random variables (probability mass function, cumulative distribution function, mean and variance). Discrete probability distributions (binomial, geometric, hypergeometric, Poisson, uniform).
12. Continuous random variables (probability density function, distrubution function, mean, variance, quantiles). Exponencial distribution. Normal distribution. Central limit theorem.
13. Testing of statistical hypotheses (t-test).

Aims

The main goal of the course is to explain the basic principles and methods of higher mathematics and probability and statistics that are necessary for the study at Brno University of Technology. The practical aspects of application of these methods and their use in solving concrete problems are emphasized.

Specification of controlled education, way of implementation and compensation for absences

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.

Classification of course in study plans

• Programme EID_P Master's, 1. year of study, winter semester, 4 credits, compulsory
• Programme REI_P Master's, 1. year of study, winter semester, 4 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Exercise

26 hours, compulsory

Teacher / Lecturer