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：

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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。