In this article, we introduce a notable bounded distribution based on a modification of the epsilon function which creates an upper bound on the domain of the distribution. Further, a key feature of the distribution links the readers with the asymptotic connections with the famous Lindley distribution, which is a weighted variant of the exponential distribution and also a mixture of exponential and gamma distributions. In some ways, the proposed distribution provides a flexible solution to the modeling of bounded characteristics that can be almost well-fitted by the Lindley distribution if the domain is restricted. Moreover, we have also explored its application, particularly with reference to lifetime and environmental points of view, and found that the proposed model exhibits a better fit among the competing models. Further, from the annual rainfall analysis, the proposed model exhibits a realistic return period of the rainfall.