I'm interested in finding the expectation values for the following scenario (physically this corresponds to the situation where you have shot noise but with low statistics).
Suppose I have a number of bins ($N_{\rm bin}$) and in them I randomly place $N_{\rm balls}$. What are the expectation values between different bins? Is this a well-defined and well-studied distribution?
In the limit where $N_{\rm balls}$ is sufficiently large I can assume the deviations from the mean of different bins are uncorrelated and so we can use a Poisson distribution (where $N_i$ denotes the number of balls in bin $i$): $$\langle (N_{A} - \bar{N})(N_{B} - \bar{N})\rangle =\delta _{AB} \langle (N-\bar{N})^2 \rangle =\delta _{AB} \bar{N}$$ but what happens as the total number of balls becomes smaller and smaller such that the trials are no longer independent (i.e., if I find a ball in the first bin there is a much smaller chance there is also one in the second bin)?