Course detail
Probability and statistics
FAST-GA03Acad. year: 2023/2024
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
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.
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)
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)
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)
Classification of course in study plans
Type of course unit
Lecture
26 hod., 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 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
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.