# Incrementally adding to a uniform distribution of samples

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?

• I am not exactly sure about your question - it seems like you would like to generate, e.g. a spatial Poisson process. Or make it simple, for a given region, you generate a Poisson random variable in which the mean is density times the area/volume of that region to give the number of objects inside. The location of each object will be given by independent uniform distribution over that region. – BGM Apr 16 '16 at 18:54
• Thanks. After looking at its description, it may be exactly what I want. I'll work this into the simulation model and see what happens. I'll come back once I have more info. – Jim Apr 17 '16 at 6:15
• @BGM I looked in to this and then experimented enough with your suggestion to believe it is exactly what I needed. If you post your comment as an answer, I will accept it. – Jim Apr 20 '16 at 15:02

and my comment is almost the same as the first paragraph in page 12: For a given region with area $A$, generate $\text{Poisson}(DA)$ to be the number of points inside this region, and then we generate independent uniform distribution over the region for the locations of each point inside.