Continuous Mixture of Uniforms Framework for Modeling Residual Lifetime Distribution

In this paper, a novel mathematical framework based on Continuous Mixture of Uniforms (CMU) for modeling the Residual Lifetime physical phenomenon is proposed. Through an alternative and intuitively comprehensible mathematical analysis, both the mixing distribution and the Residual Lifetime Theorem as a function of the Lifetime distribution are derived, providing a novel perspective compared to existing literature. The applicability of the derived mathematical expressions is demonstrated by obtaining the probability distribution function and statistical properties of the residual lifetime when the lifetime follows various commonly used distributions, including the negative exponential, log-normal, gamma, n-th order hyper-exponential, Pareto, and Weibull distributions. Furthermore, we present new mathematical expressions for generating random numbers with the Residual Lifetime distribution when the Lifetime follows either a log-normal or Weibull distribution. The proposed CMU framework is also employed to establish explicit relationships between distribution and moments of the Residual Lifetime and of the Lifetime. By using the conditional moments of the uniform random variables that constitute the CMU, functional relationships between both raw and standardized moments of residual lifetime and standardized moments of the lifetime are derived. Finally, numerical results graphically illustrate the statistical relationships between both the distribution and moments of the lifetime, mixing distribution, and residual lifetime, providing valuable insights into their interdependencies.