On the distribution of the distance of pairs of random points from a spherical shell

Key Points

• The research aims to analyze the distance distribution between independent random points in concentric spherical shells and to explore cases involving the beta distribution.
• Examined distribution using mathematical modeling of random points in spherical shells in R d.
• Redefined distance distribution for points selected from beta distribution within spherical geometries.
• Considered special cases for distance calculations in conditions of varying dimensions d.
• Derived analytical expressions for distance distributions among points in spherical shells.
• Found specific distribution characteristics for random points in a beta distribution setup.
• Identified unique properties of distance between two points within concentric shells under specified conditions.