A numerical computing for the CDF of injective RVT with optimization analysis using Weibull distribution: Applications to stochastic heat transfer

This paper presents a novel numerical method for computing the cumulative distribution function (CDF) of one-to-one univariate Random Variable Transformations (RVTs). This method is crucial for assessing the statistical representation weight of random changes within the RVT domain, offering a practical alternative to traditional analytical methods, which are often limited in applicability. The research outlines the challenges posed by the analytical computation of the CDF for one-to-one RVTs, motivating the development of a more accessible numerical approach. This approach first verifies if the RVT function is one-to-one, then uses the bisection method to approximate its inverse of RVT function, enabling numerical evaluation of the CDF of RVT based on a governing model. The procedures are algorithmically detailed to facilitate the computational implementation. Additionally, an analytical re-evaluation generates an optimal fitted path for the numerical behavior of the CDF of RVT, using the Weibull distribution model to provide a mathematical model simulating the RVT's statistical characteristics approximately. The methodology is applied to compute the probabilistic distribution of uncertain thermal properties generated from solving a stochastic heat transfer problem, due to analyzing random variations in thermal diffusivity with a Gamma distribution as a case study to address potential measurement errors.