# Count of random points belonging to a Voronoi cell in the unit square

I am considering a Voronoi tessellation in the unit square $$V \in [0,1]^2$$ for $$G$$ points uniformly randomly distributed. Then I am considering $$N$$ other points also randomly distributed uniformly in $$V$$ (and independently of $$G$$), and I want to find an expression for the p.d.f. of the counts of these points in the Voronoi cells, as I change $$G$$ and $$N$$.

This is how I am reasoning. Because $$G\gg 1$$ in my case, I can approximate the distribution of Voronoi cell areas by the Gamma distribution $$\mathrm{Gamma}(x;\nu;\nu^{-1})$$ [Kiang, 1966], where $$x$$ is the cell area $$s$$ normalized w.r.t. the mean cell area $$\rho^{-1}$$, i.e. $$x=\rho s$$. Moreover, following [Pineda et al., 2006], I get an excellent fit for my data when $$\nu=3.575$$.

Now the harder part. In the classical paper by Gilbert [1962,p. 963], it is reasoned that the probability of a randomly-picked point in the plane to belong to a cell of area $$s$$ is $$p^*(s)=\rho s \cdot p(s)$$ where $$p(s)$$ is the p.d.f. of cell areas, i.e. $$\mathrm{Gamma}(\rho s; \nu, \nu^{-1})$$ in my case. Once again, it is convenient to work in terms of the normalized cell area $$x$$, whereby I can define the p.d.f. that given one point, it belongs to a cell of size $$x$$, by $$f(x|n=1)=x\cdot \mathrm{Gamma}(x;\nu,\nu^{-1})$$.

If everything worked out fine, the probability that $$k$$ randomly-picked points belong to the same cell of size $$x$$ would then be $$f^k(x|n=1)$$ (because of the independent drawing). Moreover, assuming that $$f(k|x)$$ - the probability that given a cell of size $$x$$, it contains $$k$$ points randomly-drawn from $$N$$ - is a $$\mathrm{Binomial}(k,N,x)$$, it would follow that $$f_k(k) = f(k|x) p(x)f^{-k}(x|n=1)$$.

This reasoning is apparently seriously wrong. But I cannot figure out where I am wrong. If then everything makes sense up to this point, what am I missing/doing that I cannot correctly fit my data?

Ok, I did not find an analytical formula for the above, except for an integral expression, obtained by marginalizing $$f(k,x)$$, since there is not a direct way to compute $$f(x|n=k)$$, that is the p.d.f. that a cell of size $$x$$ contains at least $$k$$ points. So in my case, $$f_k(k) = \int_0^G f(k|x)f(x)dx$$ provides a correct estimation for the distribution of random points in a Voronoi cell in a random tessellation of $$G$$ cells in unit square.
Note: $$f(x|k)$$ is not trivial to estimate, because it certainly relates to the probability that $$k$$ independent points are each found in a cell of size $$x$$, i.e. $$f^k(x|1)$$, but in addition, we also need to estimate the probability that such cell is the same for all $$k$$ points, or in other words, the probability that randomly picking $$k$$ points, the all fall in the same cell.