Hindawi Publishing Corporation Journal of Quality and Reliability Engineering Volume 2013, Article ID 190437, 13 pages http://dx.doi.org/10.1155/2013/190437 Research Article On the Mean Residual Life Function and Stress and Strength Analysis under Different Loss Function for Lindley Distribution Sajid Ali Department of Decision Sciences, Bocconi University, via Roenthen 1, 20136 Milan, Italy Correspondence should be addressed to Sajid Ali; sajidali.qau@hotmail.com Received 4 November 2012; Accepted 4 February 2013 Academic Editor: Shey-Huei Sheu Copyright © 2013 Sajid Ali. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Purpose. Mathematical properties of Lindley distribution are derived under diferent loss functions. Tese properties include mean residual life function, Lorenz curve, stress and strength characteristic, and their respective posterior risk via simulation scheme. Methodology. Bayesian approach is used for the reliability characteristics. Results are compared on the basis of posterior risk. Findings. Using prior information on the parameter of Lindley distribution, Bayes estimates for reliability characteristics are compared under diferent loss functions. Practical Implications. Since Lindley distribution is a mixture of gamma and exponential distribution, so Bayesian estimation of reliability characteristics will have a great implication in reliability theory. Originality. A real life application to waiting time data at the bank is also described for the developed procedures. Tis study is useful for researcher and practitioner in reliability theory. 1. Introduction Exponential distribution is frequently used as a lifetime distribution in statistics and applied areas; the Lindley distri- bution has been ignored in the literature since 1958. Lindley distribution originally developed by Lindley [1] and some classical statistic properties are investigated by Ghitany et al. [2]. Sankaran [3] introduced a discrete version of Lindley distribution known as discrete Poisson-Lindley distribution, and Ghitany and Al-Mutairi [4] described some estimation methods. Te distribution of zero-truncated Poisson-Lindley was introduced by Ghitany et al. [5] who used the distribution for modeling count data in the case where the distribution has to be adjusted for the count of missing zeros. Zamani and Ismail [6] introduced negative binomial distribution as an alternative to zero-truncated Poisson-Lindley distribution. Recently, Ghitany et al. [7] introduced a two-parameter weighted Lindley distribution and pointed that Lindley dis- tribution is particularly useful in modelling biological data from mortality studies. Te rest of the study is organized as follows. Section 2 deals with the derivation of posterior distribution using dif- ferent noninformative and informative priors. Using diferent loss functions, the Bayes estimators and their respective posterior risks are discussed in Section 3. Elicitation of hyperparameter is also discussed in Section 3. Simulation study of Bayes estimates of mean residual life and their posterior risks is performed in Section 4. Lorenz curve discussion for Lindley distribution is given in Section 5 while stress and strength reliability characteristics and simulation study under diferent loss functions is discussed/performed in Section 6. Real life application is illustrated in Section 7. Finally, Section 8 deals with a conclusion and some future remarks. 2. Likelihood Function and Posterior Distributions Te posterior distribution summarizes available probabilis- tic information on the parameters in the form of prior distribution and the sample information contained in the likelihood function. Te likelihood principle suggests that the information on the parameter should depend only on its posterior distribution. Bayesian scientist’s job is to assist the investigator to extract features of interest from the posterior distribution. In this section, we will use the Lindley model as sampling distribution mingles with noninformative priors for the derivation of posterior distribution. A random variable 