the fractional discrete weibull distribution

The two-parameter discrete weibull distribution is an important model especially in reliability studies when the data are reported on a discrete scale‎. ‎the hazard rate function of a discrete weibull distribution is monotonically increasing and decreasing‎. ‎the present paper provides a family of parametric discrete distributions which is an infinite mixture of exponentiated discrete weibull distributions‎, ‎and versatile in fitting increasing‎, ‎decreasing‎, ‎and bathtub-shaped failure rate models to different discrete life-test data‎. ‎some important distributional properties of the model such as the moments‎, ‎order statistics‎, ‎and infinite divisibility are investigated and the parameters of the distribution are estimated by the maximum likelihood method‎. ‎in addition‎, ‎a real data set is analyzed to show the effectiveness of the model‎. ‎finally we conclude the paper.