Probability and Statistics

• 26 hrs lectures
• 26 hrs exercises

• 80 pts final exam

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

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

Secondary school mathematics and selected topics from previous mathematical courses.

• Hlavičková, I., Hliněná, D.: Matematika 3. Sbírka úloh z pravděpodobnosti. VUT v Brně, 2015 (CS)
• Montgomery, D. C., Runger, G. C.: Applied Statistics and Probability for Engineers. New York: John Wiley & Sons, 2011. (EN)

1. Introduction to probability theory. Combinatorics. Axiomatic probability. Classical and statistical probability.
2. 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. Continuous probability distributions: uniform, exponencial, normal.
5.  Central limit theorem. Basic arithmetics with random variables and their influence on the parameters of probability distributions.
6. 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.
7. Introduction to statistics. Descriptive statistics. Data processing.
8. Characteristics of central tendency, variability and shape. Moments. Graphical representation of the data.
9. Estimation theory. Point estimates. Maximum likelihood method. Bayesian inference. Interval estimates.
10. Statistical hypothesis testing. One-sample and two-sample tests (t-test,  F-test).
11. Goodness-of-fit and normality tests. Test of independence.
12. Correlation analysies. Pearson's and Spearman's correlation coefficient.
13. Introduction to regression analysis. Linear regression.

1. Combinatorics

2. Classical probability. Examples using combinatorics. Statistical probability.

3. Conditional probability, dependence and independence. Multiplication and addition rules for probabilities. Total probability, Bayes' theorem.

4. Random variable (discrete and continuous), probability mass function, cumulative distribution function, probability density function. Characteristics of a random variable (mean, variance, skewness, kurtosis).

5. Discrete probability distributions: Bernoulli, binomial, hypergeometric, geometric, Poisson.

6. Continuous probability distributions: uniform, exponential, normal. Central Limit Theorem. Basic linear and non-linear operations on random variables and their effect on distribution parameters.

7. Random vector (discrete and continuous). Joint and marginal probability mass functions, cumulative distribution functions, and densities. Characteristics of a random vector (mean, variance, covariance, correlation coefficient). Dependence and independence of random variables. Multivariate normal distribution.

8. Introduction to statistics. Sample surveys. Descriptive statistics. Sorting and processing datasets.

9. Measures of location, variability, and shape; sample moments and graphical representation of data.

10. Credit test

11. Estimation theory. Point estimation of distribution parameters. Maximum likelihood method. Bayesian inference. Interval estimation of distribution parameters.

12. Statistical hypothesis testing. One-sample and two-sample tests (paired and unpaired t-tests, F-test).

13. Goodness-of-fit and normality tests. Test of independence. Correlation analysis. Pearson and Spearman correlation coefficients.

• Credit test: 20 points.
• Obtaining the credit is conditional upon scoring at least 10 points on the credit test.
• Final exam: 80 points.