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Distribution of the Distance Between Two Random Points in a Body from $$\boldsymbol{R}^{\boldsymbol{n}}$$
Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) ( IF 0.3 ) Pub Date : 2020-12-28 , DOI: 10.3103/s1068362320060023
N. G. Aharonyan , V. Khalatyan

Abstract

In the present paper a formula for calculation of the density function \(f_{\rho}(x)\) of the distance between two independent points randomly and uniformly chosen in a bounded convex body \(D\) is given. The formula permits to find an explicit form of density function \(f_{\rho}(x)\) for body with known chord length distributions. In particular, we obtain an explicit expression for \(f_{\rho}(x)\) in the case of a ball of diameter \(d\).

A simulation model is suggested to calculate empirically the cumulative distribution function of the distance between two points in a body from \(R^{n}\), where explicit form of the function is hard to obtain. In particular, simulation is performed for balls and ellipsoids in \(R^{n}\).



中文翻译:

人体中两个随机点之间的距离分布,从$$ \ boldsymbol {R} ^ {\ boldsymbol {n}} $$

摘要

在本文中,给出了计算有界凸体\(D \)中随机且均匀选择的两个独立点之间距离的密度函数\(f _ {\ rho}(x)\)的公式。该公式允许找到具有已知弦长分布的物体的密度函数\(f _ {\ rho}(x)\)的显式形式。特别地,在直径为\(d \)的球的情况下,我们获得\(f _ {\ rho}(x)\)的显式表达式。

建议建立一个仿真模型,根据经验从\(R ^ {n} \)计算体内两点之间的距离的累积分布函数,而该函数很难获得显式形式。特别地,对\(R ^ {n} \)中的球和椭球执行模拟。

更新日期:2020-12-28
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