Probability norm less than threshold in unit ball

Question

From exercise 2.4 in Elements of statistical learning, studying this solution : http://tullo.ch/static/ESL-Solutions.pdf Points $x_{i}, i=1..N$ are uniformly distributed in a p-dimensional unit ball centered at origin. How do we obtain $P(\|xi\|<r)=\frac{K r^{p}}{K}$ ? I feel it is related to Mahalanobis distance https://en.wikipedia.org/wiki/Mahalanobis_distance, can't find how to extrapolate to multivariate gaussian, and to derive this result.

Answer 1

$\|x_i\| < r$ for $r \leqslant 1$ is a $p$-dimensional ball itself. And volume of $p$-dimensional ball of radius $r$ is $K_p \cdot r^p$ (where $K_p = \frac{\pi^\frac n 2}{\Gamma\left(\frac n 2 + 1\right)}$). For uniform distribution, probability of a subset of support is proportional to it's volume, so probability of getting into ball of radius $r$ is $\frac{K_p r^p}{K \cdot 1^p}$.