I want to simulate the generation of objects over a defined area $A$, where the objects have a uniform spatial distribution, $D$ objects per unit area within $A$.
I initially thought that, for $A$ big enough, the number of objects would be either $\lfloor D\times A \rfloor$ or $\lceil D\times A \rceil$ and the sample mean would be $D\times A$.
I think I can handle this for a large area, and if my random object generation is not done "exactly correctly", it would still be good enough.
But I also want to iteratively add an area $\Delta A$ and populate that area with objects. If the area is very small, then the expected number of objects for that area might be less than $1$. So I would have a random chance that I would or would not add an object to that area. Where I am going with this question is how to address that there should also be a random chance that I might add two or more objects to that area.
If $\Delta A$ is small (but not "very small"), the expected number of objects may be a small number, like 3.2. My initial thought would be to add either 3 or 4 objects, with a 3 being a certainty and the 4th one being done by a random chance.
I think this should be made more rigorous, so that while I incrementally add some number of objects to each $\Delta A$ area, the expected distribution still meets the uniform distribution in space at the density of $D$.
So while I might have an expected number $n_o$, I want to choose a probabilty distribution that will let me determine how to randomly decide the value $n$ to actually populate any area $A$ or $\Delta A$ during any individual iteration.
I was thinking a binomial distribution might apply. But I don't know if I have defined the problem enough that a specific distribution applies, or is it still not defined well enough? How would I go about choosing a distribution that meets my spatial distribution specification?
A related and more specific question is how would I analytically determine if the pdf I choose for determining $n$ will yield a population that has an expected distribution that is spatially uniform as specified?