# Probability norm less than threshold in unit ball

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\| ? 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.

• What does it mean "normally distributed in a unit ball"? Normal distribution (except for degenerate case) has infinite support. – mihaild Apr 17 at 9:36
• Consider N data points uniformly distributed in a p-dimensional unit ball centered at the origin. – kiriloff Apr 17 at 9:38

$$\|x_i\| < r$$ 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 $$\frac{K_p r^p}{K \cdot 1^p}$$.