A novel computational framework for the expected number of real roots of stochastic functions on a given interval
Xu Yi
Abstract
We propose a new computational framework for the expected number of real roots of a stochastic function on a given interval. The classical Kac-Rice formula requires the joint density of the function and its derivative, which is often intractable. Our approach avoids this requirement entirely by introducing a cumulative expectation function. Through analysis of its absolute continuity and differential structure, we derive two complementary computational schemes: one expresses the expectation as a derivative of a variable-domain integral under weak conditions; the other yields an explicit integral representation without joint densities or variable-domain differentiation. We illustrate the method in detail for linear stochastic functions, obtaining explicit formulas for Gaussian and uniform distributions, together with several new analytical results. The framework substantially broadens the scope of problems amenable to rigorous analysis and provides a powerful tool for applications in stochastic analysis and beyond.