In this paper, we introduce a new family of distributions which extends several new half-distributions. Also, the family can be viewed as a general weighted family of distributions. Some general mathematical properties of the family are obtained, involving the quantile function, (conditional) moments, moment generating function, entropies, order statistics, record values and a bivariate extension of the family. Different estimation procedures for the family parameters are discussed. Some sub-models of the family that can accommodate various shapes for the hazard rate and density functions are given. Using two reliability data sets, the potentiality of proposed sub-models of the family is shown under the estimation procedures.