The improvement of one-parameter lifetime distributions together with Lindley, Zegdoudi, XLindley, new XLindley, XGamma, etc., has usually been popular, however including a brand new parameter to a distribution makes it better and more flexible than the present one. This paper presents a new family of two-parameter (NTPFD) derivatives of the two parameter polynomial exponential family. This new version is a generalization of numerous novel one-parameter distributions such as: the XLindley, new XLindley, and ZLindley distributions. We study the main statistical properties of NTPFD: the shape of the density and hazard rate functions, moments, skewness, kurtosis, coefficient of variation, Bonferroni and Lorenz curves, reliability parameters, stochastic ordering, entropies, and quantile function. A discussion of fuzzy reliability is also given. To study the applicability, usefulness, reliability and superiority of the proposed distribution over existing distributions, two real data sets are assessed and fitted to the NTPFD distribution and potential competitors such as: an uncensored data set corresponding to the remission duration of a random sample of 128 patients with bladder cancer and the US population recorded in the decennial census during the period 1790-1970.