Probability density function of the Cartesian x-coordinate of the random point inside the hypersphere

TL;DR

This paper derives the probability density function of the x-coordinate of a random point inside an n-dimensional hypersphere, showing its relation to the Gaussian distribution and convergence to the standard normal distribution as dimensions increase.

Contribution

It provides an explicit form of the x-coordinate's PDF inside a hypersphere and demonstrates its convergence to the Gaussian distribution in high dimensions.

Findings

The PDF is explicitly derived for the x-coordinate.

The distribution converges to the standard normal distribution as n increases.

The work links geometric probability with Gaussian behavior in high dimensions.

Abstract

Consider randomly picked points inside the n-dimensional unit hypersphere centered at the origin of the Cartesian coordinate system. The Cartesian coordinates of the points are random variables, which form an n-dimensional vector for each point. Observing only the x-coordinate I obtained its probability density function (PDF). I show that it is related to the Gaussian distribution: in limit its companion PDF?? converges to the PDF of the standard normal distribution.