I have a D-dimensional space of volume V, and I uniformly sample it $P$ times by randomly positioning points / throwing darts / etc. I also randomly position some number, $N$, of non-overlapping $D$-spheres, for the same value $D$ as the dimension of the space, with volume $v_{sphere}$. So we have circles for $D$ = 2, regular spheres for $D$ = 3, higher dimensional spheres for greater values of $D$. What is the probability for finding a certain number, $r$, of my $P$ points/darts/etc. in a given $D$-sphere?
Probability distribution for the number of points in a D-sphere when uniformly sampling a D-dimensional space
1
$\begingroup$
probability-distributions
-
1@joriki, thanks, I'll be a lot more careful next time! – 2011-07-27
1 Answers
4
I'm assuming that when you say you have a $D$-dimensional space of volume $V$, you mean a subset of volume $V$ of a $D$-dimensional Euclidean space, or at least of a space with a Euclidean metric.
The positioning and number of the spheres is irrelevant; they could be positioned randomly or fixed at arbitrary positions, and there could be any number of them. What matters is only the ratio $\rho$ of the volume of a sphere to the volume $V$ of the subset. The probability for finding $r$ of the $P$ points in a given sphere is then given by the binomial distribution: $\binom Pr\rho^r(1-\rho)^{P-r}$.
-
0@Srivatsan, sorry about that... – 2011-07-27