From the context of bird surveys (how many breeding bird territories overlap a circular sampling area), is there a more suitable distribution to use, rather than poisson?
In this case, a poisson distribution is commonly used to describe the distribution of counts over various sampling points.
My issue is that poisson represents counts of spatial points (which zero area/volume). When doing bird surveys, territories are hexagon shaped (for the sake of simplicity). The area of these hexagons provide a ceiling which is not present in a poisson distribution (the right hand tail of the poisson can represent theoretically impossible values).
In my mind, a better suited distribution would be derived from a spatial map of adjacent hexagons, and while randomly placing circles on this space (representing observers), the Probability Density Function would represent the distribution of the number of hexagons which overlaps each of the randomly placed observer circles. The relative size of the hexagons vs the size of observer sampling area would describe the mean of this distribution.
Any thoughts preexisting probability density functions which may address this need? In my mind (which is somewhat uninformed about statistics), if this would be combined with a binomial distribution which represents the occupancy of the said territories, this would be a more suitable tool for species distribution modelling.
Indeed, it seems that perhaps I should move this onto the math forum, as there seems to be closely related questions on there, as I'm more interested in the proof of the distribution rather than the practical aspects of calculating how well alternate distributions fit this situation. I'm thinking that there are no commonly used distributions for this purpose, and I'll have to write my own probability density function based on geometry proofs.