Probabilistic modeling of random variables with inconsistent data

The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.