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.

翻译：本文研究了牛顿法算法在特定情形下输出的统计特性。该样本的相对频率密度收敛于一个定义明确的函数，这促使我们探究其分布规律。通过严格的数学证明，我们论证了其概率密度函数服从柯西分布。此外，本文提出了一种生成均匀分布的新方法。为验证结论，我们采用包括Kolmogorov-Smirnov检验与Anderson-Darling检验在内的统计测试，结果显示较高的p值。进一步地，我们证明通过将变换方法应用于柯西分布，可获得连续两次输出之间距离的分布。