Extensions of the Freund Distribution with Applications in Reliability Theory

The Freund distribution models situations in which the exponential residual lifetime of one component depends on the working status of another component. The literature has discussed generalizations and analogs of the Freund distribution. Such generalizations can be obtained by replacing exponential random variables by Weibull or gamma random variables, or by compounding mixing Freund distributions. This paper shows how to unify various well-known generalizations of the Freund distribution by using one simple functional representation. This functional representation has a physical meaning that sheds new light on the meaning of some of these generalizations. Furthermore, using this functional representation, we derive various properties of the Freund distribution and its extensions. In particular, we show that some random variables, which have the Freund distribution or one of its extensions, are positively associated and also belong to some classes of multivariate "increasing failure rate average" and "new better than used" distributions. These properties provide bounds for various probabilistic quantities of interest in reliability theory.