Probability and Statistics

• 26 hrs lectures
• 26 hrs exercises

• 70 pts final exam (written part)
• 30 pts numeric exercises

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

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.

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.

• Discrete Mathematics (IDM)
• Calculus 1 (IMA1)
• Calculus 2 (IMA2)

Secondary school mathematics and selected topics from previous mathematical courses.

• 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)
• 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)

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.

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

Exam prerequisites:
Get at least 10 points during the semester.

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

Get at least 10 points during the semester.