Variance of x_i chosen from uniformly distributed hypersphere I'm looking for an expression of the variance of a single component of a point chosen from within a uniformly distributed n-ball with radius r for any n.
There are a few proofs showing that components of a point chosen this way become increasingly normally distributed as n tends to infinity. I can't seem to find expressions for any n.

I didn't totally follow @kimchi lover's answer (particularly his assumption of normality), so here is a similar approach.
Suppose X is hyperspherically uniformly distributed with r=1, so each $X_i$ has $E[X_i] = 0$, and we want to compute $VAR[X_i]$. Since $E[X_i]=0$, by the variance-squares identity, we have $VAR[X_i] = E[X^{2}_i] - E[X_i]^2 = E[X^{2}_i]$, so we really just need to compute $E[X^{2}_i]$.
Notice that $r^2 = \Sigma_{i=1}^n X^{2}_i$, so we have $E[r^2] = nE[X^{2}_i]$. Since we know that r has pdf $p_n (r) = nr^{n-1}$, we can trivially compute $E[r^2] = \int_0^1 r^2 nr^{n-1}dr = \frac{n}{n+2}$, so then $E[X^{2}_i] = VAR[X_i] = \frac{E[r^2]}{n} = \frac{1}{n+2}$.
