Journal of Mathematical Sciences, Vol.218, No. 3, October, 2016 THE WEIBULL–RAYLEIGH DISTRIBUTION, SOME PROPERTIES, AND APPLICATIONS M. Ganji 1 , H. Bevrani 2 , N. Hami Golzar 1 , and S. Zabihi 2 A new distribution, the Weibull–Rayleigh distribution, is introduced, and various properties of the distribution are provided. Two real data sets are used to illustrate the applicability of the Weibull– Rayleigh distribution. 1. Introduction The Weibull distribution has been widely used in lifetime studying, reliability, and fitting a best model to data. To add flexibility to the Weibull distribution, various generalizations of the distribution have been derived, including the generalized Weibull distribution [10], the exponentiated Weibull family [11], and the beta-Weibull distribution by [9]. Let Y be an exponential random variable with the probability density function f (y)= e -y ,y> 0; then by the transformation technique T = γY 1 c + m, γ> 0,c> 0, is a Weibull random variable with location parameter m, shape parameter c, and scale parameter γ . The density function of T is r(t)= c γ  t - m γ  c-1 exp  -  t - m γ  c  , t  m, c, γ > 0. (1) If m = 0 then the density function and the distribution function of T are respectively given as r(t)= c γ  t γ  c-1 exp  -  t γ  c  , γ> 0,c> 0, (2) R(t)=1 - exp  -  t γ  c  , γ> 0,c> 0. (3) The Rayleigh distribution is also applicable in clinical studies, bioassay, etc. This distribution was introduced by Rayleigh [8]. Let X be a Rayleigh random variable with parameters b; then the density function and the distribution function of X are respectively expressed as f (x)= x b 2 exp  - x 2 2b 2  ,x  0,b> 0, (4) 1 Department of Statistics, University of Mohaghegh Ardabili, Arbadil, Iran, e-mail: mganji@uma.ac.ir , nasrin.hami@uma.ac.ir 2 Department of Statistics, University of Tabriz, Tabriz, Iran, e-mail: bevrani@tabrizu.ac.ir , soheila.zabihi@yahoo.com Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014. 1072-3374/16/2183-0269 2016 Springer Science+Business Media New York 269 DOI 10.1007/s10958-016-3028-2