Uniform on unit sphere

If we draw uniformly form the unit sphere, and call its coordinates $(X,Y,Z)$, what is the variance of $X$?

Solution

This seems hard to calculate explicitly, but looking at the symmetry of the problem yields an easy answer. We know that by construction every coordinate is mean 0 and that

\[X^2 + Y^2 + Z^2 = 1\]

and by symmetry $\mathrm{Var}(X) = \mathrm{Var}(Y) = \mathrm{Var}(Z)$. Therefore, we must have $\mathrm{Var}(X) = 1/3$.