This article puts forth a novel category of probability distributions obtained from the Topp–Leone distribution, the inverse- exponential distribution, and the power functions. To obtain this new family, we used the original cumulative distribution functions. After introducing this new family, we gave the motivations that led us to this end and the basis of the new family obtained, followed by the mathematical properties related to the family. Then, we presented the statistic order, the quantile function, the series expansion, the moments, and the entropy (Shannon, Reiny, and Tsallis), and we estimated the parameters by the maximum likelihood method. Finally, using real data, we presented numerical results through data analysis with a comparison of rival models.