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?

Apparently the answer is

$$ f_X(x) = \frac{1}{2R} $$

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.

  • $\begingroup$ 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. $\endgroup$
    – whuber
    Aug 12, 2020 at 19:56
  • $\begingroup$ Also asked at math.stackexchange.com/q/3788624/321264 $\endgroup$ Aug 12, 2020 at 20:02


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