In this paper, the mixture of two components of exponentiated family is introduced as a new family of continuous distributions. Some general properties of the proposed family are discussed such as the quantile function, moments, moment generating function and order statistics. The maximum likelihood estimation method is used to derive the estimators for the unknown parameters, reliability and hazard rate functions. The mixture of two exponentiated inverted Kumaraswamy distribution is studied as a sub-model from the mixture exponentiated family of distributions. Some statistical properties are studied such as the quantile function, moments, moment generating function and order statistics. Also, the maximum likelihood estimators for the unknown parameters, reliability and hazard rate functions of the mixture exponentiated inverted Kumaraswamy distribution are obtained. A simulation study is conducted to assess the performance of the maximum likelihood estimators of the parameters of the mixture exponentiated inverted Kumaraswamy distribution. Finally, a real data set is applied to ensure the simulated results.