Calculation of quantiles of a uniform distribution over a sphere

Question

How do we calculate quantiles of a uniform distribution over a sphere ? Can anyone provide me with a tutorial ?

Answer 1

The question needs some interpretation. I will assume that we have a uniform distribution over a ball of radius $c$. If $P$ is a "randomly chosen" point in the ball, let $X$ be the distance of $P$ from the centre of the ball. We are interested in the various quantiles of $X$.

For example, let us compute the $60$-th percentile of $X$. This is the number $x=x_{0.6}$ such that $\Pr(X\le x)=0.6$. The probability that $X$ is $\le x$ is the volume of the ball of radius $x$ divided by the volume of the whole ball. This ratio is $\frac{x^3}{c^3}$. Thus we want $$\frac{x^3}{c^3}=0.6.$$ Solve. We get $$x=x_{0.6}=c\sqrt[3]{0.6}.$$ Any other quantile that we want is computed in the same way.