# How to find Poisson Point Process in a fixed rectangle?

I want to simulate a problem with MATLAB,but I must first find two-dimensional Poisson Points Process in a rectangle $[0,1] \times [-n,n]$, that $n$ is known! I don't have any information about its density $\lambda$, these points should be Poisson Points Process of any $\lambda$. I read this A program for spatial point generation using Poisson processes

In its 2nd and 3rd page it describes that the procedure should be the following:

The procedures for the realisation of homogeneous Poisson process in a two dimensional region $\mathcal{R}$ are given as:

1. Sub-divide the region $\mathcal{R}$ into $m$ sub-regions, $A_1, A_2, …, A_m$. In most of the cases, region $\mathcal{R}$ is a rectangle and $A_1, A_2, …, A_m$ are disjoint rectangular quadrates with $\mathcal{R} = \bigcup_{i} A_i$. Work out the mean of the Poisson distribution for each quadrate as $\mu_i = \lambda_i \nu(A_i)$ , where $\lambda_i$ is the density for quadrate $i$ and $\nu(A_i)$ is the area of $A_i$. For homogeneous case, $\lambda_1 = \lambda_2 = ... = \lambda_m = \lambda$.

2. For each of the $m$ sub-regions $A_i$, generate a random variable $N_i$ based on the Poisson distribution function.

3. For each of the n events for quadrate $A_i$, generate two random variables according to the uniform distribution and use them as the locations inside the sub-region.

4. Repeat procedures 2 and 3 until all quadrates in $\mathcal{R}$ are visited

In my problem I just want to define $f$ on two-dimensional Poisson process points. if $x = (x_1,x_2)$ is a poisson point process and its projection is $y = (y_1,y_2)$ that we have for them:

$$x_1 < y_1,\quad x_2 - 1 \leq y_2 \leq x_2 + 1$$

And there isn't any point except two points $x , y$ in $[x_1,y_1] \times [x_2 - 1,x_2 + 1]$.It can be shown that with starting from one of the points $x$ or $y$ there exists natural numbers $m$ and $n$ that $f^m(x) = f^n(y)$.Now with this information again I don't have any idea how to choose $\lambda$.And is this homogeneous or non homogeneous problem?

A Poisson process simply means that the probability a point is placed in any given square is the same, like raindrops falling onto floor tiles. Well-known proofs show that the distribution of points in squares (the number of squares with $0$ points, with $1$ point, with $2$ points, etc.) then obeys the Poisson distribution formula.
But once you generate your points, you might form a histogram of the resulting set of points and fit it with a Poisson Distribution with mean $\mu$ and see how well they agree.
• You mean that I pick a square from $[0,1] \times [-n,n]$ at random and then I add a random point in this square? Commented Jul 20, 2017 at 20:42