Exponential-modified discrete Lindley distribution View Article: PubMed Central - PubMed ABSTRACTIn this study, we have considered a series system composed of stochastically independent M-component where M is a random variable having the zero truncated modified discrete Lindley distribution. This distribution is newly introduced by transforming on original parameter. The properties of the distribution of the lifetime of above system have been examined under the given circumstances and also parameters of this new lifetime distribution are estimated by using moments, maximum likelihood and EM-algorithm. No MeSH data available. © Copyright Policy - OpenAccess Related In: Results  -  Collection License getmorefigures.php?uid=PMC5037113&req=5 .flowplayer { width: px; height: px; } Fig2: P.d.f of EMDL random variable for different parameter values Mentions: Thus, we can obtain the marginal probability density function of X as3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( x;\theta ,\beta \right) =\frac{{\theta }^2}{1+2\theta} \frac{\beta e^{-\beta x}\left( 3-\left( 1-\theta \right) e^{-\beta x}\right) }{{\left( 1-\left( 1-\theta \right) e^{-\beta x }\right)}^3},\quad x>0$$\end{document}fx;θ,β=θ21+2θβe-βx3-1-θe-βx1-1-θe-βx3,x>0where θ ∈ (0, 1) and β > 0. Henceforth, the distribution of the random variable X having the p.d.f in (3) is called shortly EMDL. By changing of variables in cumulative integration of (3), the distribution function can be found as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\left( x;\theta ,\beta \right) =1-\left[ \frac{{\theta }^2}{1+2\theta }\frac{e^{-\beta x}\left( 3-2\left( 1-\theta \right) e^{-\beta x}\right) }{{\left( 1-\left( 1-\theta \right) e^{-\beta x}\right) }^2}\right].$$\end{document}Fx;θ,β=1-θ21+2θe-βx3-21-θe-βx1-1-θe-βx2.Following figure shows different shapes of p.d.f of EMDL random variable for various values of and (Fig. 2).Fig. 2

Exponential-modified discrete Lindley distribution
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Mentions: Thus, we can obtain the marginal probability density function of X as3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left( x;\theta ,\beta \right) =\frac{{\theta }^2}{1+2\theta} \frac{\beta e^{-\beta x}\left( 3-\left( 1-\theta \right) e^{-\beta x}\right) }{{\left( 1-\left( 1-\theta \right) e^{-\beta x }\right)}^3},\quad x>0$$\end{document}fx;θ,β=θ21+2θβe-βx3-1-θe-βx1-1-θe-βx3,x>0where θ ∈ (0, 1) and β > 0. Henceforth, the distribution of the random variable X having the p.d.f in (3) is called shortly EMDL. By changing of variables in cumulative integration of (3), the distribution function can be found as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\left( x;\theta ,\beta \right) =1-\left[ \frac{{\theta }^2}{1+2\theta }\frac{e^{-\beta x}\left( 3-2\left( 1-\theta \right) e^{-\beta x}\right) }{{\left( 1-\left( 1-\theta \right) e^{-\beta x}\right) }^2}\right].$$\end{document}Fx;θ,β=1-θ21+2θe-βx3-21-θe-βx1-1-θe-βx2.Following figure shows different shapes of p.d.f of EMDL random variable for various values of and (Fig. 2).Fig. 2