# A novel computational framework for the expected number of real roots of stochastic functions on a given interval

## TL;DR

This paper introduces a new computational framework to estimate the expected number of real roots of stochastic functions, bypassing the intractable joint density calculations of classical methods, and provides explicit formulas for specific distributions.

## Contribution

A novel framework that avoids joint density calculations by using a cumulative expectation function, enabling analysis of stochastic functions' roots with broader applicability.

## Key findings

- Explicit formulas for Gaussian and uniform distributions.
- New analytical results for linear stochastic functions.
- Broader applicability of root expectation analysis.

## 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.