  In this paper we propose a new four-parameters distribution with increasing,
decreasing, bathtub-shaped and unimodal failure rate, called as the
exponentiated Weibull-Poisson (EWP) distribution. The new distribution arises
on a latent complementary risk problem base and is obtained by compounding
exponentiated Weibull (EW) and Poisson distributions. This distribution
contains several lifetime sub-models such as: generalized exponential-Poisson
(GEP), complementary Weibull-Poisson (CWP), complementary exponential-Poisson
(CEP), exponentiated Rayleigh-Poisson (ERP) and Rayleigh-Poisson (RP)
distributions.
  We obtain several properties of the new distribution such as its probability
density function, its reliability and failure rate functions, quantiles and
moments. The maximum likelihood estimation procedure via a EM-algorithm is
presented in this paper. Sub-models of the EWP distribution are studied in
details. In the end, Applications to two real data sets are given to show the
flexibility and potentiality of the new distribution.
