Methods and algorithms for the class of probability distributions based on a combination of Meier G-functions and Fox H-functions and their applications

PhD study program: Applied Mathematics
Akademic year: 2020-2021
Advisor: doc. RNDr. Viktor Witkovský, CSc. (witkovsky@savba.sk)
External educational institution: Institute of Measurement Science SAS
Accepting university: Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Department of Applied Mathematics and Statistics

Annotation:
The class of G-functions and the class of H-functions were designed by Meijer and Fox as a generalization of a broad class of mathematical functions which with different parameters cover almost all known elementary and special functions (e.g. exponential function, gamma function, hypergeometric function, and Bessel functions). In addition, these functions have many good mathematical properties that can be advantageously used for their manipulation (e.g., the Mellin transform, the Laplace transform, and the Fourier transform). It turns out that density (PDF) respectively the cumulative distribution function (CDF) of many probability distributions can be expressed using these special functions (as e.g. gamma distribution, beta distribution, chi-square distribution, F-distribution). It is a very broad class of probability distributions that find applications in various fields of natural, technical and biomedical sciences. It follows from the above properties of these functions that the distribution of a combination of independent random variables with this type of distribution can also be expressed by simple manipulation of these functions. On the other hand, there are obviously many probability distributions that can be expressed by G-functions or H-functions that have not yet been identified, but may be useful for modeling complex physical or biological processes. The PhD thesis has several important objectives. The main objective is focused on research of advanced methods for measurement evaluation and development of methods and algorithms for computing probability distributions based on combinations of Meier G-functions and Fox H-functions and their applications. The specific objective of the project is the characterization of probability distributions expressible by G-functions and H-functions and development of efficient numerical methods and algorithms for computing the distributions of algebraic functions of independent random variables which can be made by using these distributions.

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