M. Ghorbani - Department of mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
S.F Bagheri - Department of Mathematics, College of Basic Science, Yadegar–e-Imam Khomeini (RAH) Branch, Islamic Azad University, Tehran, Iran
M. Alizadeh - Department of Mathematics, College of Basic Science, Yadegar–e-Imam Khomeini (RAH) Branch, Islamic Azad University, Tehran, Iran

The Weibull distribution has been shown to be useful for modeling and analysis of life time data in medical, biological and engineering sciences. We proposed a new four-parameters distribution with decreasing, increasing and bathtub-shaped failure rate, called as the modified Weibull Poisson distribution. The new distribution arises on a latent complementary risk problem base and is obtained by compounding modified Weibull and Poisson distributions. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival, failure rate functions, the r-th raw moment of the new distribution and also the moments of order statistics, expressions for the Rényi and Shannon entropies. Moreover, we also discuss maximum likelihood estimation and provide formulae for the elements of the Fisher information matrix. Finally, applications of two real data sets is given to show the flexibility and potentiality of the new distribution. We hope that this generalization may attract wider applications in reliability and survival analysis.

Modified Weibull distribution, Compounding, Lifetime distributions, Maximum likelihood estimation, Residual life function