  In this paper a new lifetime distribution, which is called the exponentiated
Weibull-geometric (EWG) distribution, is introduced. This new distribution
obtained by compounding the exponentiated Weibull and geometric distributions.
The EWG distribution includes as special cases the generalized
exponential-geometric (GEG), complementary Weibull-geometric (CWG),
complementary exponential-geometric (CEG), exponentiated Rayleigh-geometric
(ERG) and Rayleigh-geometric (RG) distributions.
  The hazard function of the EWG distribution can be decreasing, increasing,
bathtub-shaped and unimodal among others. Several properties of the EWG
distribution such as quantiles and moments, maximum likelihood estimation
procedure via an EM-algorithm, R\'{e}nyi and Shannon entropies, moments of
order statistics, residual life function and probability weighted moments are
studied in this paper. In the end, we give two applications with real data sets
to show the flexibility of the new distribution.
