# Why is the marginal pdf of $x$ constant? $x$ is the coordinate of the points uniformly distributed on the surface of a sphere [duplicate]

Consider a sphere of radius $$R$$ centered at the origin with points uniformly distributed on the surface. What is the marginal pdf of the x-coordinate of these points?

$$f_X(x) = \frac{1}{2R}$$
This is very unintuitive for me because I would think this $$f_X$$ would be dependent on $$x$$, and would be largest at $$x = 0$$ tapering off as x increases/decreases.
• The marginal is the distribution of the dot product of the coordinates with the unit x vector $(1,0,0),$ which is why the answer is found in one of the duplicates concerning scalar products. This result is correct for the sphere in three dimensions only: it underlies the most basic equal-area maps of the earth.