Exploring the Statistical Properties of Outputs from a Process Inspired by Geometrical Interpretation of Newton's Method

TL;DR

This paper investigates the statistical distribution of outputs from a geometrically inspired Newton's method, proving it follows a Cauchy distribution and proposing a new uniform distribution generation method.

Contribution

It provides a rigorous proof that the output distribution is Cauchy and introduces a novel method for generating uniform distributions.

Findings

Output distribution converges to a Cauchy distribution.

Statistical tests confirm the Cauchy distribution fit.

A transformation method for distances between outputs is developed.

Abstract

In this paper, the statistical properties of Newton s method algorithm output in a specific case have been studied. The relative frequency density of this sample converges to a well-defined function, prompting us to explore its distribution. Through rigorous mathematical proof, we demonstrate that the probability density function follows a Cauchy distribution. Additionally, a new method to generate a uniform distribution is proposed. To further confirm our findings, we employed statistical tests, including the Kolmogorov-Smirnov test and Anderson-Darling test, which showed high p-values. Furthermore, we show that the distribution of the distance between two successive outputs can be obtained through a transformation method applied to the Cauchy distribution.