Folding Domain Functions (FDF): a Random Variable Transformation technique for the non-invertible case, with applications to RDEs

TL;DR

This paper introduces a novel, computationally efficient method for determining probability distributions of random variables transformed by non-invertible functions, with applications to random differential equations and discrete mappings.

Contribution

It presents a new approach to the Random Variable Transformation method that handles non-invertible functions, expanding its applicability and ease of implementation.

Findings

Method is easy to implement and computationally low-cost.

Successfully applied to examples involving random differential equations.

Effective for non-bijective transformations in applied physics contexts.

Abstract

The Random Variable Transformation (RVT) method is a fundamental tool for determining the probability distribution function associated with a Random Variable (RV) Y=g(X), where X is a RV and g is a suitable transformation. In the usual applications of this method, one has to evaluate the derivative of the inverse of g. This can be a straightforward procedure when g is invertible, while difficulties may arise when g is non-invertible. The RVT method has received a great deal of attention in the recent years, because of its crucial relevance in many applications. In the present work we introduce a new approach which allows to determine the probability density function of the RV Y=g(X), when g is non-invertible due to its non-bijective nature. The main interest of our approach is that it can be easily implemented, from the numerical point of view, but mostly because of its low computational cost, which makes it very competitive. As a proof of concept, we apply our method to some numerical examples related to random differential equations, as well as discrete mappings, all of them of interest in the domain of applied Physics.