Certain Properties and Applications of Generalized Gamma and Beta Distributions

Special functions frequently arise in many areas of mathematics, physics, and engineering. In this research paper, we consider two well-known special functions: the generalized gamma function Γkh and the generalized beta function βkg,h. We investigate two important extensions of the classical probability distributions, namely the k-gamma and k-beta distributions. These distributions are generalizations of the classical gamma and beta distributions, respectively, incorporating an additional parameter k, which provides greater flexibility for modeling real-life data. We derive and analyze explicit formulas for the mean and variance of both distributions. We also figure out the survival and hazard rate functions, for these two distributions. This helps us understand how reliable they are. The survival and hazard rate functions are important because they show us how the distributions behave when it comes to reliability. We use the survival and hazard rate functions of the distributions to see how well they work. The results illustrate how the parameter k influences the behavior of these distributions. Such generalizations are particularly useful in reliability theory and engineering applications, where classical distributions may not adequately fit observed data.