In this paper we introduce a generalization of the Bilal distribution, where a new two–parameter distribution is presented. We show that its failure rate function can be upside-down bathtub shaped. The failure rate can also be decreasing or increasing. A comprehensive mathematical treatment of the new distribution is provided. The estimation by maximum likelihood is discussed; and a closed–form expression for Fisher’s information matrix is obtained. Asymptotic interval estimators for both of the two unknown parameters are also given. A simulation study is conducted and applications to real data sets are presented.