ABSTRACT
This chapter describes some specific univariate distributions that are directly related to the exponential distribution. Mixtures of exponential distributions are often used to represent variation from a pure exponential distribution in models of lifetime distributions. They correspond to the concept of a population composed of several subpopulations, within each of which there is a constant hazard rate. Exponential distributions are a special case of gamma distributions - they belong to the gamma family. Exponential distributions, therefore, belong to the Pearson Type III class of distributions, and also a fortiori to the broader natural exponential family. K. Ryu extended the bivariate exponential distribution of Marshall and Olkin to an absolutely continuous model, not necessarily possessing the memoryless-property. Based on the statistical theory of extreme values and the Griffith theory of brittle fracture, C. M. Black et al., derived the brittle fracture distributions which include exponential and Weibull distributions as special cases.
