@guest Sorry for the slow response, I was away from my computer. However that's a good suggestion, and I'll do just that.

The heart of the question here is - if we sample random variates from the distribution, as a function of the distribution's dimensions (with respect to the dimensions of an underlying integer lattice), how well can we estimate the distribution's center? Unfortunately, I don't have some actual physical system in mind here.

First of all, thanks for reading my question. Secondly, I did mean what I said - that there is a unique fixed probability that a query of the particle's position fails depending on the coordinate that it would otherwise have been binned to. However, I feel that this part of the question unnecessarily complicates things, and isn't as defined as it should be, so I've gotten rid of it.

@LiviuNicolaescu I will also think about how to make the question formulation clearer - thanks for bearing with me.

@LiviuNicolaescu I have hopefully clarified $C$ a bit better now, and I provide an explicit example for calculating $C^*$ on a $3 \times 3$ matrix (is there a better technical term for what I'm doing?). I should have known better about using the term "elliptic" function.

@LiviuNicolaescu The example was meant as a simplified one-dimensional illustration of what I meant by "pixel value weighted average". In the actual problem, one needs to take into consideration both $x$ and $y$-components. Let me clarify the example.

@LiviuNicolaescu Sorry about that, I just meant to say that it should be an average weighted by the values corresponding to the pixels.