| Abstract: | Different properties of the HNBUE (HNWUE) class of life distributions (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx\, \le \,(\ge)\,\mu }\]$\end{document} exp(?t/μ) for t ≥ 0, where μ = \documentclass{article}\pagestyle{empty}\begin{document}$\int_t^\infty {\,\,\,\mathop F\limits^-(x)\,dx}$\end{document} are presented. For instance we characterize the HNBUE (HNWUE) property by using the Laplace transform and present some bounds on the survival function of a HNBUE (HNWUE) life distribution. We also examine whether the HNBUE (HNWUE) property is preserved under the reliability operations (i) formation of coherent structure, (ii) convolution and (iii) mixture. The class of distributions with the discrete HNBUE (discrete HNWUE) property (i.e.), for which \documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=k}^\infty {\mathop{\mathop P\limits^-_{j\,\,\,}\, \le(\ge)\,\mu(1 - 1/\mu)^{^k }}\limits^{}} $\end{document} for k = 0, 1, 2, ?, where μ =\documentclass{article}\pagestyle{empty}\begin{document}$\sum\limits_{j=0}^\infty {\mathop {\mathop P\limits^- _{j\,\,\,\,\,}and\mathop P\limits^ - _{j\,\,\,\,\,}=}\limits^{}}\,\,\sum\limits_{k=j+1}^\infty {P_k)}$\end{document} is also studied. |
