All the M–O distributions mentioned above and the GLE distribution are special cases of the new family that is introduced and studied in Section 2. In Section 3, some statistical and reliability properties of the new family are discussed. In Section 4, the method of maximum likelihood estimation is used to estimate the unknown parameters. In addition, simulation is utilized to calculate the unknown parameters and to study their properties. Section 5 gives some applications to explain how some real data sets can be modeled by the new family. Finally, in Section 6, some conclusions and remarks of the current and future research are presented.