Course details

# Probability and Statistics

IPT Acad. year 2019/2020 Winter semester 5 credits

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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

Fusek Michal, Ing., Ph.D. (DMAT FEEC BUT)

Deputy Guarantor

Language of instruction

Czech

Completion

Credit+Examination (written)

Time span

26 hrs lectures, 26 hrs pc labs

Assessment points

70 exam, 30 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.

Prerequisites

Prerequisite kwnowledge and skills

Secondary school mathematics and selected topics from previous mathematical courses.

Study literature

• Anděl, J.: Statistické metody. Praha: Matfyzpress, 1993. (CS)
• Anděl, J.: Základy matematické statistiky. Praha: Matfyzpress, 2005. (CS)
• 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)
• Likeš, J., Machek, J.: Matematická statistika. Praha: SNTL - Nakladatelství technické literatury, 1988. (CS)
• Neubauer, J., Sedlačík, M., Kříž, O.: Základy statistiky. Praha: Grada Publishing, 2012. (CS)

Fundamental literature

• 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)
• 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)

Syllabus of lectures

1. Introduction to probability theory. Combinatorics and classical probability.
2. Axiomatic probability. Conditional probability and independence. Probability rules. Total probability, Bayes' theorem.
3. Random variable (discrete and continuous), probability mass function, cumulative distribution function, probability density function. Characteristics of random variables (mean, variance, skewness, kurtosis).
4. Discrete probability distributions: Bernoulli, binomial, hypergeometric, geometric, Poisson.
5. Continuous probability distributions: uniform, exponencial,  normal. Central limit theorem.
6. Basic arithmetics with random variables and their influence on the parameters of probability distributions.
7. 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.
8. Descriptive statistics. Data processing. Characteristics of central tendency, variability and shape. Moments. Graphical representation of the data.
9. Random sample. Point estimates. Maximum likelihood method.
10. Interval estimates. Statistical hypothesis testing. One-sample and two-sample tests (t-test,  F-test).
11. Goodness-of-fit test. Analysis of variance (ANOVA). One-way and two-way ANOVA.
12. Correlation and regression analyses. Linear regression. Pearson's and Spearman's correlation coefficient.
13. Bayesian statistics. Conjugate prior. Maximum a posteriori probability (MAP) estimate. Posterior predictive distribution.

Syllabus of numerical exercises

Practising of selected topics of lectures.

Progress assessment

• Written tests: 30 points.
• Final exam: 70 points.

Controlled instruction

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

Exam prerequisites

Get at least 10 points during the semester.

Course inclusion in study plans

• Programme BIT, 2nd year of study, Compulsory
• Programme IT-BC-3, field BIT, 2nd year of study, Compulsory