0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What does it mean "normally distributed in a unit ball"? Normal distribution (except for degenerate case) has infinite support. $\endgroup$ – mihaild Apr 17 at 9:36
  • $\begingroup$ Consider N data points uniformly distributed in a p-dimensional unit ball centered at the origin. $\endgroup$ – kiriloff Apr 17 at 9:38
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.