Poisson Point Pattern

Q. If we randomly generate x and y coordinates in 2D using the uniform distribution (typical random number generator), what will the distribution of xy-coordinates be?

A. “Poisson” distribution

Proof of concept: A coordinate point is considered as an event happened in the area W. Let’s say we have a grid of 20x20 cells. The 2,000 randomly generated coordinates in the grid is shown as follows:

It does not look uniform now, does it? If we count the number of points in each cell, we can have the histogram of them as:

The density is the number of points per cell area. For example, in this case, it is 2000/(20x20) = 5.

We can calculate the mean and variance of the number of points in each cell (statisticians call it ‘quadrat’) and also the ratio of variance to mean. From the above example, we got these numbers:

var = 4.9824,  mean=5, VMR=0.9965 $\approx $1

What type of distribution has its mean equal to its variance?  Poisson distribution, of course (Its mean = variance = $\lambda$).

Actually, the above histogram is in a form (realization) of Poisson distribution with parameter $\lambda=5$ or $P(X) = \frac{{{e^{ - 5}}{5^x}}}{{X!}}$ where X=0,1,2,3,... points in a cell. It is also an example of spatial Poisson process.