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Exponential-modified discrete Lindley distribution

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ABSTRACT

In 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.

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Mrl function of EMDL random variable for different parameter values
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Fig5: Mrl function of EMDL random variable for different parameter values

Mentions: The mean residual life function of X is given by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} mrl\left( x\right)&=E\left( X-x\vert {X>x}\right) \ \\&=\frac{1}{\beta }\ \frac{r\left( 1-r\right) -2{\left( 1-r\right) }^2ln\left( 1-r\right) }{3r-2r^2} \end{aligned}$$\end{document}mrlx=EX-x/X>x=1βr1-r-21-r2ln1-r3r-2r2where . Note that holds for . We can see this result immediately below by letting . Then applying the mean value theorem, we have the upper bound for as . If this upper bound is written above, then\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$mrl\left( x\right) \le \frac{1}{\beta }\ \frac{3\left( 1-r\right) }{3-2r}\le \frac{1}{\beta }.$$\end{document}mrlx≤1β31-r3-2r≤1β.We have the following graphs of mrl(x) for different values of parameter and (Fig. 5).Fig. 5


Exponential-modified discrete Lindley distribution
Mrl function of EMDL random variable for different parameter values
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Fig5: Mrl function of EMDL random variable for different parameter values
Mentions: The mean residual life function of X is given by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} mrl\left( x\right)&=E\left( X-x\vert {X>x}\right) \ \\&=\frac{1}{\beta }\ \frac{r\left( 1-r\right) -2{\left( 1-r\right) }^2ln\left( 1-r\right) }{3r-2r^2} \end{aligned}$$\end{document}mrlx=EX-x/X>x=1βr1-r-21-r2ln1-r3r-2r2where . Note that holds for . We can see this result immediately below by letting . Then applying the mean value theorem, we have the upper bound for as . If this upper bound is written above, then\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$mrl\left( x\right) \le \frac{1}{\beta }\ \frac{3\left( 1-r\right) }{3-2r}\le \frac{1}{\beta }.$$\end{document}mrlx≤1β31-r3-2r≤1β.We have the following graphs of mrl(x) for different values of parameter and (Fig. 5).Fig. 5

View Article: PubMed Central - PubMed

ABSTRACT

In 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.