If we have points uniformly distributed over the volume of a sphere, what is the distribution of their x coordinate?

Question

My understanding is that if we just have points uniformly distributed on the surface then they’re uniformly distributed along any given axis, struggling to generalise to the volume case though

Answer 1

this is intended to be a comment not an answer ... but again ... wanted to add graphics ... if you run a Monte Carlo (in this case 10000 points) and keep only those within a certain radius, histogram of x-coordinate looks as follows $$ $$

in this case x-, y- and z-coordinates were randomly generated in [0,1] and then only those within radius $\frac{1}{2}$ of $\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ were retained ... again ... not intended to be an answer

Answer 2

For points uniformly distributed over the volume, and $x \in [-R,R]$ on the axis, the distribution will be proportional to $a (R^2-x^2)$where $a$ is a constant to be computed based on the total density of points in the volume of the sphere.

The idea is that the density of points is proportional to the area of the disk obtained by intersecting the volume of the sphere and the plane orthogonal to the axis at the given point. And the radius of the disk is given by Pythagore’s theorem.