Probability Theory II
- Credits: 5
- Ending: Examination
- Range: 2P + 2C
- Semester: winter
- Year: 2
- Faculty of Economic Informatics
Teachers
Included in study programs
Teaching results
By completing the course Probability Theory II, students will expand their knowledge of one-dimensional probability distributions of random variables. They will gain knowledge about the apparatus for calculating probabilities and numerical characteristics (even conditional) in the case of two-dimensional distributions. They can use in the second part of the course, in which they will deal with the issue of determining the joint distribution using a copula functions. This appropriately captures the possible dependence of marginal random variables. Using R language software support, resp. MS Excel will also be able to simulate values from these distributions, which they can use in solving various problems in the field of actuarial services, such as in the aggregation of risks.
Knowledge
Students will gain knowledge about two-dimensional probability distributions, their characteristics, and the use of generating values from these distributions. They will also gain knowledge about determining the distribution of the sum of random variables. At the same time, they will have knowledge of the problem of modeling two-dimensional distributions of random variables using a copula functions.
Competences
Based on the acquired knowledge, students will have the competence to select a suitable approach for the implementation of probabilistic calculations, as well as to assess the dependence of random variables. They will be able to evaluate the selection of a suitable copula function for modeling data from a data file.
Skills
After completing the course, students can:
• perform probabilistic calculations,
• determine the dependence between random variables,
• implement and use the generation of values of two-dimensional random variables in solving
problems,
• implement various graphic outputs and interpretations,
• use computer technology and software support (R language, MS Excel),
• implement aggregation of random variables using copula functions.
Indicative content
1. The concept of multidimensional random variable. Marginal distribution of variables. Joint probability mass function and joint cumulative distribution function of a discrete two-dimensional random variable.
2. Joint probability density function and joint cumulative distribution function of a continuous two-dimensional random variable.
3. Laws of conditional distributions of two-dimensional random variable. Dependence of random variables, covariance and correlation. Graphic interpretation.
4. Numerical characteristics of conditional distributions of a two-dimensional random variable.
5. Generating functions of marginal random variables, convolution, Laplace transform.
6. Distribution of the sum of two marginal random variables and its characteristics.
7. Two-dimensional normal and t distribution. Generating their values. Visualization in the R language environment.
8. Copula functions. Properties of copula functions. Sklar's theorem. Survival copula.
9. Dependency measures (Pearson correlation coefficient, Kendall and Spearman rank correlation coefficient) and tail dependence.
10. Classification of copula functions. Elementary copulas (independent, comonotonic), Implicit copulas(Gaussian, Student's), Archimedean copulas(Clayton's, Frank's, Gumbel's) and others.
11. Calibration of the copula function. Parametric and nonparametric estimation of parameters. Verification of the selection of a suitable copula function.
12. Simulation of copula functions, generation of two-dimensional distribution values using copula functions, their visualization using scatterplot.
13. Use of copula functions in aggregation of two marginal random variables.
Support literature
1. Fecenko, J.: Teória pravdepodobnosti II v Maxime. Letra Edu. 2018.
2. Ruppert, D., Matteson S., D.: Statistics and Data Analysis for Financial Engineering with R examples. Springer. 2015.
3. Everitt, B., Hothorn, T.: An Introduction to Applied Multivariate Analysis with R. Springer. 2011.
4. Hofert, M., Kojadinovic, I., Mächler, M., & Yan,J.: Elements of copula modeling with R. Springer. 2018.
5. Devore, L.,J.: Probability & Statistics for Engineering and the Sciences. Brooks/Cole. 2012.
6. Charpentier, A.: Computation actuarial science with R. Taylor & Francis Group. 2015.
7. Joe, H.: Dependence Modeling with Copulas. Taylor & Francis Group, LLC. 2015.
8. Škrovánková, L., Simonka, Z. Aktuárske metódy a modely v penzijnom, zdravotnom a nemocenskom poistení. Brno : H.R.G., 2021.
Syllabus
1. The concept of multidimensional random variable. Marginal distribution of variables. Joint probability mass function and joint cumulative distribution function of a discrete two-dimensional random variable. 2. Joint probability density function and joint cumulative distribution function of a continuous two-dimensional random variable. 3. Laws of conditional distributions of two-dimensional random variable. Dependence of random variables, covariance and correlation. Graphic interpretation. 4. Numerical characteristics of conditional distributions of a two-dimensional random variable. 5. Generating functions of marginal random variables, convolution, Laplace transform. 6. Distribution of the sum of two marginal random variables and its characteristics. 7. Two-dimensional normal and t distribution. Generating their values. Visualization in the R language environment. 8. Copula functions. Properties of copula functions. Sklar's theorem. Survival copula. 9. Dependency measures (Pearson correlation coefficient, Kendall and Spearman rank correlation coefficient) and tail dependence. 10. Classification of copula functions. Elementary copulas (independent, comonotonic), Implicit copulas(Gaussian, Student's), Archimedean copulas(Clayton's, Frank's, Gumbel's) and others. 11. Calibration of the copula function. Parametric and nonparametric estimation of parameters. Verification of the selection of a suitable copula function. 12. Simulation of copula functions, generation of two-dimensional distribution values using copula functions, their visualization using scatterplot. 13. Use of copula functions in aggregation of two marginal random variables.
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,
16 hours - preparation for exercises, homeworks,
20 hours - preparation for written works,
42 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