We study a new family of distributions defined by the minimum of the Poisson random number of independent and
identically distributed random variables having the Topp Leone-G distribution. Some mathematical properties of the new
family are derived. Maximum likelihood estimation of the model parameters is investigated. Two special models of the new
family are discussed. We perform three applications to real data sets to show the potentiality of the proposed family. In order
to test the validity of the new family, a modified Chi-squared goodness-of-fit test based on Nikulin-Rao-Robson statistics is
proposed theoretically.