Consider two independent point processes in the unit square $[0,1]^2$. The two point processes are identically independent and typically binomial/Poisson. One, say $\Phi^*$, is used to generate a Voronoi tessellation of $N^*$ cells. The other one, $\Phi$ with $N$ points, is used to build the statistics of the counts of points of $\Phi$ in the Voronoi cells associated with $\Phi^*,N^*$.

Gilbert's argument (Ann. Math. Stat., 1962) for a randomly picked given point in plane to be contained in a Voronoi cell of area $s$ is that, known the p.d.f. of cell areas -- be it $f(s)$ --, then the probability of $f(s|X=1)=sE[s]^{-1}f(s)$ where $X$ is a random variable that expresses the count of points in $s$. How can I generalize the argument to any $X$ value? In principle, if I consider $k$ points randomly drawn and independently, I thought of considering $f^k(s|X=1)$, but it does not match simulations. Any suggestion?