# Probability Theory I

- Credits: 7
- Ending: Examination
- Range: 2P + 2C
- Semester: winter
- Year: 2
- Faculty of Economic Informatics

## Teachers

## Included in study programs

**Teaching results**

After completing the course Probability Theory I, it is assumed that students will acquire knowledge and skills in the field of probability distributions, which can be used in a stochastic approach to problem solving. Also thanks to the software support of the R language and an innovative approach in the form of simulations of the values of a random variable, a better and deeper understanding of the meaning of various numerical characteristics and their interpretations, and clearer approach to mastering the probabilistic apparatus. This creates the potential for future stochastic analyzes of data that are used in various sectors of practice. The acquired knowledge, competencies and skills are a basic prerequisite for subsequent education in the field of statistics.

Knowledge

Students will gain knowledge of concepts and rules within the calculation of probability in connection with the theory of random events, they will be able to interpret the calculated probability based on its statistical definition. It is assumed that they can handle the issue of a random variable in the context of its probability distribution and numerical characteristics. They will also gain knowledge of selected discrete and continuous probability distributions used to solve problems in practice. The prerequisite is also the control of the meaning of the law of large numbers, determining the accuracy of the estimation of theoretical probability using relative frequency based on limit theorems. Emphasis is placed on understanding the meaning and interpretation of these findings and the apparatus used from the point of view of probability theory, in connection with the real acquisition of values of a random variable.

Competences

Based on the above knowledge, students are able to solve problems based on a stochastic approach within the new acquired competencies. To achieve relevant results, they can choose a probability distribution that appropriately describes the assigned task. Based on graphical interpretations of the analyzed distribution by means of the probability density function and frequency histogram are competent to decide on its important characteristics. In this context, they think about creative comparisons through creative thinking, for example within the theory of mean and variance. The acquired knowledge enables them to interpret the determined numerical characteristics with the needs of analytical practice, for example in the case of quantiles to present these values not only graphically, but also in the context of the percentage acquisition of values of a random variable. In connection with the law of large numbers and limit theorems, they are able to comment on the issue of the implementation of repeated independent experiments in connection with the estimation of the probability of occurrence of the observed event.

Skills

Within the software support of the R language, they will acquire certain skills also in this environment, while to obtain the required outputs they use prepared source codes and rewrite only those parameters which are marked in bold. Other skills include the implementation of simulations of random variable values from selected discrete and continuous distributions used in practice, as well as skills in creating frequency histograms and verification based on the processing of such generated values. Important skills are the implementation of various probability calculations to determine the probabilities and numerical characteristics, and in addition to verbal, especially their graphical interpretation, not only using functions available in the environment in the R language.

**Indicative content**

1. The probability of a random event.

2. Addition and multiplication of probabilities, conditional probability.

3. Repeated independent and dependent experiments.

4. Discrete random variable.

5. Continuous random variable.

6. Generation of values of discrete and continuous random variable.

7. Numerical characteristics of a discrete random variable.

8. Numerical characteristics of a continuous random variable.

9. Discrete distributions: binomial, geometric, negative binomial distribution.

10. Discrete distributions. hypergeometric, Poisson distribution, approximations.

11. Continuous distributions: uniform distribution, exponential, gamma distribution and others.

12. Continuous distributions: normal and normed normal distribution.

13. Law of large numbers, central limit theorems.

**Support literature**

1. Mucha,V., Páleš, M.: Teória pravdepodobnosti pre ekonómov. S podporou jazyka R. Letra Edu. 2018.

2. Horáková, G., Huťka, V.: Teória pravdepodobnosti. Ekonóm. 2010.

3. Dobrow, R.: Probability: With Applications and R. John Wiley & Sons. 2014.

4. Horgan, J.: Probability with R. An Introduction with Computer Science Applications. John Wiley & Sons. 2009.

**Syllabus**

1. The probability of a random event. 2. Addition and multiplication of probabilities, conditional probability. 3. Repeated independent and dependent experiments. 4. Discrete random variable. 5. Continuous random variable. 6. Generation of values of discrete and continuous random variable. 7. Numerical characteristics of a discrete random variable. 8. Numerical characteristics of a continuous random variable. 9. Discrete distributions: binomial, geometric, negative binomial distribution. 10. Discrete distributions. hypergeometric, Poisson distribution, approximations. 11. Continuous distributions: uniform distribution, exponential, gamma distribution and others. 12. Continuous distributions: normal and normed normal distribution. 13. Law of large numbers, central limit theorems.

**Requirements to complete the course**

30% 2 written works (using software support),

70% written exam (using software support)

**Student workload**

Total study load (in hours): 156 hours

26 hours - participation in lectures,

26 hours - participation in exercises,

26 hours - preparation for exercises, homeworks,

20 hours - preparation for written works,

58 hours - self-study in preparation for the exam.

**Language whose command is required to complete the course**

slovak

Date of approval: 10.02.2023

Date of the latest change: 15.05.2022