Beta-polynomial a priori densities in bayesian reliability models

The Bayesian approach to constructing models of the reliability theory is considered. Within this approach, the model is considered to be incomplete in a certain sense — it is assumed that the key parameters of the system are random variables with known a priori distributions. At some time points, the modifications are introduced to the system to improve reliability; however, each modification may either increase or reduce the reliability of the system. Thus, system's reliability characteristics depend on the ratio of the modification means' parameters of “efficiency” to the parameters of “defectiveness.” Such relation can be called the “system's balance index.” In this paper, the case of beta-polynomial a priori distributions is considered, where one of the parameters of the system has an a priori beta distribution and the density of the other parameter has the form of a polynomial. For various combinations of given a priori distributions, the formulas for calculating the probabilistic characteristics of the balance index are provided.