Course details

# Probability and Statistics

IPT Acad. year 2022/2023 Winter semester 5 credits

Classical probability. Axiomatic probability. Conditional probability. Total probability. Bayes' theorem. Random variable and random vector. Characteristics of random variables and vectors. Discrete and continuous probability distributions. Central limit theorem. Transformation of random variables. Independence. Multivariate normal distribution. Descriptive statistics. Random sample. Point and interval estimates. Maximum likelihood method. Statistical hypothesis testing. Goodness-of-fit test. Analysis of variance. Correlation and regression analyses. Bayesian statistics.

Guarantor

Course coordinator

Language of instruction

Completion

Time span

- 26 hrs lectures
- 26 hrs exercises

Assessment points

- 80 pts final exam
- 20 pts numeric exercises

Department

Lecturer

Instructor

Subject specific learning outcomes and competences

Acquired knowledge can be applied, for example, in other courses or in the BSc/MSc thesis.

Learning objectives

The main goal of the course is to introduce basic principles and methods of probability and mathematical statistics which are useful not only in computer sciences.

Why is the course taught

The world around us is complex and we try to describe it using mathematical models. However, not all the models are deterministic. In many situations, randomness plays an important role and some phenomena occur only with a certain probability. Students will learn about the probability, learn how to model the behaviour of random variables and analyze obtained (measured) data using selected methods of mathematical statistics. Overall, probability and related topics are important parts of computer science.

Recommended prerequisites

- Discrete Mathematics (IDM)
- Mathematical Analysis 1 (IMA1)
- Mathematical Analysis 2 (IMA2)

Prerequisite knowledge and skills

Secondary school mathematics and selected topics from previous mathematical courses.

Study literature

- Fajmon, B., Hlavičková, I., Novák, M., Vítovec, J.: Numerická matematika a pravděpodobnost (Informační technologie), VUT v Brně, 2016 (CS)
- Hlavičková, I., Hliněná, D.: Matematika 3. Sbírka úloh z pravděpodobnosti. VUT v Brně, 2015 (CS)
- Anděl, J.: Matematická statistika. Praha: SNTL, 1978. (CS)
- Anděl, J.: Statistické metody. Praha: Matfyzpress, 1993. (CS)
- Anděl, J.: Základy matematické statistiky. Praha: Matfyzpress, 2005. (CS)
- Hlavičková, I., Hliněná, D.: Matematika 3. Sbírka úloh z pravděpodobnosti. VUT v Brně, 2015 (CS)
- Casella, G., Berger, R. L.: Statistical Inference. Pacific Grove, CA: Duxbury Press, 2001. (EN)
- Hogg, R. V., McKean, J., Craig, A. T.: Introduction to Mathematical Statistics. Boston: Pearson Education, 2013. (EN)
- Likeš, J., Machek, J.: Matematická statistika. Praha: SNTL - Nakladatelství technické literatury, 1988. (CS)
- Montgomery, D. C., Runger, G. C.: Applied Statistics and Probability for Engineers. New York: John Wiley & Sons, 2011. (EN)
- Neubauer, J., Sedlačík, M., Kříž, O.: Základy statistiky. Praha: Grada Publishing, 2012. (CS)
- Montgomery, D. C., Runger, G. C.: Applied Statistics and Probability for Engineers. New York: John Wiley & Sons, 2011. (EN)

Syllabus of lectures

- Introduction to probability theory. Combinatorics and classical probability.
- Axiomatic probability. Conditional probability and independence. Probability rules. Total probability, Bayes' theorem.
- Random variable (discrete and continuous), probability mass function, cumulative distribution function, probability density function. Characteristics of random variables (mean, variance, skewness, kurtosis).
- Discrete probability distributions: Bernoulli, binomial, hypergeometric, geometric, Poisson.
- Continuous probability distributions: uniform, exponencial, normal. Central limit theorem.
- Basic arithmetics with random variables and their influence on the parameters of probability distributions.
- Random vector (discrete and continuous). Joint and marginal probability mass function, cumulative distribution function, probability density function. Characteristics of random vectors (mean, variance, covariance, correlation coefficient). Independence. Multivariate normal distribution.
- Introduction to statistics. Descriptive statistics. Data processing. Characteristics of central tendency, variability and shape. Moments. Graphical representation of the data.
- Estimation theory. Point estimates. Maximum likelihood method. Bayesian inference.
- Interval estimates. Statistical hypothesis testing. One-sample and two-sample tests (t-test, F-test).
- Goodness-of-fit tests.
- Introduction to regression analysis. Linear regression.
- Correlation analysies. Pearson's and Spearman's correlation coefficient.

Syllabus of numerical exercises

Practising of selected topics of lectures.

Progress assessment

- Homeworks: 20 points.
- Final exam: 80 points.

Controlled instruction

Class attendance. If students are absent due to medical reasons, they should contact their lecturer.

Exam prerequisites

Get at least 8 points during the semester.

Course inclusion in study plans

- Programme BIT, 2nd year of study, Compulsory
- Programme BIT (in English), 2nd year of study, Compulsory
- Programme IT-BC-3, field BIT, 2nd year of study, Compulsory