Let $\mu$ be some distribution (with density) on $\mathbb{R}^d$, from which we independently draw $X\_1,\ldots,X\_n$. These induce a Voronoi partition on $\mathbb{R}^d$: $V\_1$ is the set of all points closest to $X\_1$, and so forth till $V\_n$ (since $\mu$ has a density, ties will almost surely not occur). I am interested in the random variable $Y\_i=\mu(V\_i)$. From symmetry, we have $E[Y\_i]=1/n$. On the line, one can show that $Y\_i$ is distributed exponentially with parameter $n$ (this is known for $\mu$ uniform over $[0,1]$ but I think a simple argument extends this to arbitrary $\mu$). What is known about the distribution of $Y\_i$ in $\mathbb{R}^d$ for general $\mu$?



Measure of random Voronoi cell
See 


*Rathie, P. N.*, [**On the volume distribution of the typical Poisson-Delaunay cell**](http://dx.doi.org/10.2307/3214909), J. Appl. Probab. 29, No. 3, 740-744 (1992). [ZBL0768.52014](https://zbmath.org/?q=an:0768.52014).


(summary - it's complicated but tractable).


