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Imagine that we have a particle sampling positions on a two-dimensional plane according to a bivariate probability distribution: $A*e^{-(\frac{(x-x_0)}{2\sigma_x^2}+\frac{(y-y_0)}{2\sigma_y^2})}$, with an unknown mean $\mu = (x_0,y_0)$ where $x_0$ and $y_0$ are real number coordinates.

Consider the problem of estimating $\mu$ by drawing $N$ iid random real number variates from the probability distribution (corresponding to real number two-dimensional coordinates), then rounding these variates to the nearest integer coordinate. For example, we may draw a real valued coordinate $(10.49344..., 2.553244)$ according to the described PDF, and we would then report this coordinate as $(10, 3)$.

Let $N$ be the number of sampling events. As a function of the $\sigma_x \approx \sigma_y$, i.e. the standard deviation parameters for our bivariate probability distribution for the particle's position, and the real number valued mean of the particle's position, $\mu$, how well can we estimate $\mu$?

Of course, if $\sigma_x$ and $\sigma_y$ are $<<1$, and depending on the difference between $\mu$ and the nearest integer coordinate, the error can be quite significant for small $N$. I'm curious how the required value of $N$ scales for localizing $\mu$ with some error $\epsilon$.

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  • $\begingroup$ I think you have a typo in part (2) where you need to get rid of one of 'failing' or '1-' or 'no'. Your problem sounds suspiciously applied, so maybe a more applied stackexchange would give better answers. For example on mathoverflow we are pedantic so we will say things like if all p(i,j) are 'always fails to report' then you won't be able to estimate mu very well. $\endgroup$
    – guest
    Dec 1, 2013 at 20:17
  • $\begingroup$ 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. $\endgroup$
    – Richard
    Dec 1, 2013 at 21:37
  • $\begingroup$ 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. $\endgroup$
    – Richard
    Dec 1, 2013 at 21:38
  • $\begingroup$ Just a suggestion if you want to get even more at the heart of the problem you could skip the part about the times and just say that the samples are independent. $\endgroup$
    – guest
    Dec 1, 2013 at 22:30
  • $\begingroup$ @guest Sorry for the slow response, I was away from my computer. However that's a good suggestion, and I'll do just that. $\endgroup$
    – Richard
    Dec 2, 2013 at 15:04

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