Where to use Poisson point process/ Uniform distribution When modeling the random locations of points in a one by one square, I am free to either model them by Poisson point process or Uniform distribution:
1) $N$ points are located in a one by one square according to a uniform distribution
2) points are located in a one by one square according to a Poisson point process with mean $N$
After modeling, I discuss the problem for large $N$, ($N \to \infty$). When should I choose Poisson and when should I choose uniform?
I know it depends on the problem I am considering, and I have to see to which of the distributions (Poisson vs Uniform) is the real problem is close to. But, if I know nothing about the real problem and have the freedom to choose either Poisson or Uniform, how should I choose?
 A: In your uniform distribution of $N$ points in a unit square: 


*

*there will be exactly $N$ points in the entire square (so $0$ variance)

*in a subset of the square of area $A$, the number of points will be binomially distributed with parameters $N$ and $A$ (so mean $NA$ and variance $NA(1-A)$) 

*in a second, mutually distinct, subset of the square with area $B$, the number of points will again be binomially distributed, this time with parameters $N$ and $B$; the number of points in the two areas will be negatively correlated
In your Poisson point process in a unit square with mean $N$: 


*

*the number of points in the unit square will have a Poisson distribution with parameter $N$ (i.e. the mean and variance)

*in a subset of the square of area $A$, the number of points will will have a Poisson distribution with parameter $NA$ 

*in a second, mutually distinct, subset of the square with area $B$, the number of points will will have a Poisson distribution with parameter $NB$; the number of points in the two areas will be independent 
You can choose whichever you think more suitable for your model.  If you choose the Poisson point process model and then observe the total number $n$, the conditional distributions in areas $A$ and $B$ given $n$ in total become the same as in the unconditional uniform distribution model with that value of $N=n$  
A: The content of the question has changed since this was posted, 
It points are distributed in the square according to a Poisson process, then the conditional distribution of the locations of the sites, given the number of sites, is the same as if they were an i.i.d. sample from a uniform distribution in the square.
So that uniform distribution is appropriate when you are given the number of sites in the square. If the number of sites is uncertain, then you have a Poisson process.
