Course detail

Probability and statistics

FAST-GA03Acad. year: 2024/2025

Random experiment, continuous and discrete random variable (vector), probability function, density function, probability, cumulative distribution, transformation of random variables, marginal distribution, independent random variables, numeric characteristics of random variables and vectors, special distributions.
Random sampling, statistic, point estimation of distribution parameter, desirable properties of an estimator, confidence interval for distribution parameter, fundamentals for hypothesis testing, tests of hypotheses for distribution parameters, goodness-of-fit test.

Language of instruction

Czech

Number of ECTS credits

3

Mode of study

Not applicable.

Guarantor

RNDr. Helena Koutková, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

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, knowledge of their basic applications.

Rules for evaluation and completion of the course

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

Aims

After the course, the students should undertand the basics of the theory of probability, work with distribution functions, know the meanig and methods of calculation of basic numeric characteristics of random variables and vectors, know how a normal random variable is defined and what is its principal meaning, know how to calculate the probability in special cases of discrete and continuous diostribution laws, know how to determine the distribution of a transformed random variable.
They should be able to interpret the basic concepts of the mathematical statistics - sampling, point estimates of distribution parameters and the reqiured properties of an estimate. They should know what an interval estimate of a distribution parameter is and be able to calculate such inerval estimates of the parameters of a normal random variable. They should know the basics of the testing of statistical hypotheses, know how to test hypotheses on the parameters of a normal random variable and on the shape of a distribution law.
Student will be able to solve simple practical probability problems and to use basic statistical methods from the fields of interval estimates,and testing parametric and non-parametric statistical hypotheses.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

KOUTKOVÁ, H., DLOUHÝ, O. Sbírka příkladů z pravděpodobnosti a matematické statistiky. Brno: CERM,2011, 63 s. ISBN 978-80-7204-740-6. (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., MOLL, I. Základy pravděpodobnosti. Brno: CERM, 2011, 127 s. ISBN 978-80-7204-738-3. (CS)
KOUTKOVÁ, H.  M03 Základy teorie odhadu a M04 Základy testování hypotéz. FAST VUT, Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp] (CS)

Recommended reading

ANDĚL, J. Statistické metody. Praha: MatFyzPress, 2007, 299 s. ISBN 80-7378-003-8.  (CS)
WALPOLE, R.E., MYERS, R.H. Probability and Statistics for Engineers and Scientists. New York: Macmillan Publishing Company, 1990, 823 p. ISBN 0-02-946910-4. (EN)

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Helena Koutková, CSc.

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 of random variable. Marginal random vector and its distribution. 4. Independent random variables. Numeric characteristics of random variable: mean and variance, quantiles. Rules of calculation mean and variance. 5. Numeric characteristics of random vectors: covariance, correlation coefficient. Normal distribution - definition, using. 6. Chi-square distribution, Student´s distribution. Random sampling, sample statistics. 7. Point estimation of distribution parameters, desirable properties of an estimator - definition, interpretation. 8. Confidence interval for distribution parameters. 9. Fundamentals of hypothesis testing. Tests of hypotheses for normal distribution parameters. 10. Goodness-of-fit tests.

Exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Helena Koutková, CSc.

Syllabus

1. Empirical distributions. Histogram. Probability and density distributions. 2. Probability. Cumulative distribution. 3. Relationships between probability, density and cumulative distributions. 4. Transformation of random variable. 5. Calculation of mean, variance and quantiles of random variable. Calculation rules of mean and variance. 6. Correlation coefficient. Calculation of probability in some cases of discrete probability distributions - alternative, binomial, Poisson. 7. Calculation of probability for normal distribution. Work with statistical tables. Calculation of point estimators. 8. Confidence interval for normal distribution parameters. 9. Tests of hypotheses for normal distribution parameters. 10. Goodness-of-fit tests.