Course Details
Mathematics 4
Academic Year 2024/25
BA004 course is not part of any programme in the faculty
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.
Credits
5 credits
Language of instruction
Czech, English
Semester
winter
Course Guarantor
Institute
Forms and criteria of assessment
course-unit credit and examination
Entry Knowledge
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.
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.
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.
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.
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)
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)
Recommended Reading
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)
Offered to foreign students
Not to offer
Course on BUT site
Lecture
13 weeks, 2 hours/week, elective
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
13 weeks, 2 hours/week, compulsory
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.