Conditional Distributions and Reliability Analysis Associated with Marshall-Olkin Bivariate Exponential Distribution

Marshall-Olkin bivariate exponential distribution

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

By considering the bivariate exponential distribution due to Marshall and Olkin (1967), we derive the conditional distributions and regression equations. It is observed that the regression equations are non-linear. An alternative derivation for the moment generating function using three independent exponential distributions is also given. Assuming that the joint distribution of the
component failure times is bivariate exponential, we obtain the reliability measures for two-unit standby, parallel and series systems.