Course detail

# Mathematics 4

Discrete and continuous random variable and vector, probability function, density function, probability, cumulative distribution, transformation of random variables, independence of random variables, numeric characteristics of random variables and vectors, special distribution laws. Random sample, point estimation of an unknown distribution parameter and its properties, interval estimation of a distribution parameter, testing of statistical hypotheses, tests of distribution parameters, goodness-of-fit tests, basics of regression analysis.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Student will be able to solve simple practical probability problems and to use basic statistical methods from the fields of interval estimates, testing parametric and non-parametric statistical hypotheses, and linear models.

Prerequisites

Basics of the theory of one- and more-functions (derivative, partial derivative, limit and continuous functions, graphs of functions). Calculation of definite integrals, double and triple integrals, knowledge of their basic applications.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Continuous and discrete random variable (vector), probability function, density function. Probability.
2. Properties of probability. Cumulative distribution and its properties.
3. Relationships between probability, density and cumulative distributions. Marginal random vector.
4. Independent random variables. Numeric characteristics of random variables: mean and variance, standard deviation, variation coefficient, modus, quantiles. Rules of calculation mean and variance.
5. Numeric characteristics of random vectors: covariance, correlation coefficient, covariance and correlation matrices.
6. Some discrete distributions - discrete uniform, alternative, binomial, Poisson - definition, using.
7. Some continuous distributions - continuous uniform, exponential, normal, multivariate normal - definition applications.
8. Chi-square distribution, Student´s distribution - definition, using. Random sampling, sample statistics.
9. Distribution of sample statistics. Point estimation of distribution parameters, desirable properties of an estimator.
10. Confidence interval for distribution parameters.
11. Fundamentals of hypothesis testing. Tests of hypotheses for normal distribution parameters.
12. Goodness-of-fit tests. Chi - square test. Basics of regression analysis.
13. Linear model.

Work placements

Not applicable.

Aims

The students should get an overview of teh basic properties of probability to be able to deal with simple practical problems in probability. They should get familiar with the basic statistical methods used for interval estimates, testing statistical hypotheses, and linear model.

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

KOUTKOVÁ, H., MOLL, I. Základy pravděpodobnosti. CERM, 2011. 127 s. ISBN 978-80-7204-783-3.  (CS)
KOUTKOVÁ, H. Základy teorie odhadu. Brno: CERM, 2007. 51 s. ISBN 978-80-7204-527-3.   (CS)
KOUTKOVÁ, H. Základy testování hypotéz. Brno: CERM, 2007. 52 s. ISBN 978-80-7204-528-0.   (CS)
KOUTKOVÁ, H., DLOUHY, O. Sbírka příkladů z pravděpodobnosti a matematické statistiky. CERM Brno, 2011. ISBN 978-80-7204-740-6.  (CS)
KOUTKOVÁ, H. Elektronické studijní opory. M03 - Základy teroie odhadu, M04 - Základy testování hypotéz. FAST VUT Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp ]   (CS)

WALPOLE, R.E., MYERS, R.H. Probability and Statistics for Engineers and Scientists. 8th ed. London: Prentice Hall, Pearson education LTD, 2007.823 p. ISBN 0-13-204767-5.  (EN)

Classification of course in study plans

• Programme B-P-C-MI (N) Bachelor's

branch MI , 2. year of study, winter semester, compulsory

• Programme B-P-E-SI (N) Bachelor's

branch S , 3. year of study, winter semester, compulsory

• Programme B-P-C-SI (N) Bachelor's

branch S , 3. year of study, winter semester, compulsory

• Programme B-K-C-SI (N) Bachelor's

branch S , 3. year of study, winter semester, compulsory
branch M , 3. year of study, winter semester, compulsory

• Programme B-P-C-SI (N) Bachelor's

branch M , 3. year of study, winter semester, compulsory

• Programme B-P-E-SI (N) Bachelor's

branch M , 3. year of study, winter semester, compulsory

• Programme B-P-C-SI (N) Bachelor's

branch N , 3. year of study, winter semester, compulsory
branch K , 3. year of study, winter semester, compulsory

• Programme B-K-C-SI (N) Bachelor's

branch K , 3. year of study, winter semester, compulsory

• Programme B-P-E-SI (N) Bachelor's

branch K , 3. year of study, winter semester, compulsory
branch V , 3. year of study, winter semester, compulsory

• Programme B-P-C-SI (N) Bachelor's

branch V , 3. year of study, winter semester, compulsory

• Programme B-K-C-SI (N) Bachelor's

branch V , 3. year of study, winter semester, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

1. Continuous and discrete random variable (vector), probability function, density function. Probability. 2. Properties of probability. Cumulative distribution and its properties. 3. Relationships between probability, density and cumulative distributions. Marginal random vector. 4. Independent random variables. Numeric characteristics of random variables: mean and variance, standard deviation, variation coefficient, modus, quantiles. Rules of calculation mean and variance. 5. Numeric characteristics of random vectors: covariance, correlation coefficient, covariance and correlation matrices. 6. Some discrete distributions - discrete uniform, alternative, binomial, Poisson - definition, using. 7. Some continuous distributions - continuous uniform, exponential, normal, multivariate normal - definition applications. 8. Chi-square distribution, Student´s distribution - definition, using. Random sampling, sample statistics. 9. Distribution of sample statistics. Point estimation of distribution parameters, desirable properties of an estimator. 10. Confidence interval for distribution parameters. 11. Fundamentals of hypothesis testing. Tests of hypotheses for normal distribution parameters. 12. Goodness-of-fit tests. Chi - square test. Basics of regression analysis. 13. Linear model.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. Empirical probability and density distributions. Histogram. 2. Probability and density distributions. Probability. 3. Cumulative distribution. Relationships between probability, density and cumulative distributions. 4. Transformation of random variable. 5. Marginal and simultaneous random vector. Independence of random variables. 6. Calculation of mean, variance, standard deviation, variation coefficient, modus and quantiles of a random variable. Calculation rules of mean and variance. 7. Correlation coefficient. Test. 8. Calculation of probability in some cases of discrete probability distributions - alternative, binomial, Poisson. 9. Calculation of probability for normal distribution. Work with statistical tables. 10. Calculation of sample statistics. Application problems for their distribution. 11. Confidence interval for normal distribution parameters. 12. Tests of hypotheses for normal distribution parameters. 13. Goodness-of-fit test.