"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th point." I was reading this in some paper.
My question is what is PPP? Of course I look for it in google and I found that means Poisson Point Process. I read the first definition of it in Wikipedia page but I still do not get it.
What is the intensity $\lambda$? Why we cannot simply say that these points are randomly located at $X_j$ with Poisson distribution?
What I understand is this: These points are in fact generated randomly with poisson distribution but if one picks a random region, then the density (or the number) of the point will be approximatively equals $\lambda$ in average. Is my understanding right?
Any help please? Thank you very much for your time.
Best Answer
In general, a point process is a random variable $N$ from some probability space $(\Omega,\mathcal{F},P)$ to a space of counting measures on ${\bf R}$, say $(M,\mathcal{M})$. So each $N(\omega)$ is a measure which gives mass to points $$ \ldots < X_{-2}(\omega) < X_{-1}(\omega) < X_0(\omega) < X_1(\omega) < X_2(\omega) < \ldots $$ of ${\bf R}$ (here the convention is that $X_0 \leq 0$. The $X_i$ are random variables themselves, called the points of $N$.
The intensity of a point process is defined to be $$ \lambda_N = {\bf E}[N(0,1]]. $$
There are many different possible point processes, but the Poisson point process with intensity $\lambda$ is the one for which the number of points in an interval $(0,t]$ has a Poisson distribution with parameter $\lambda t$: $$ P[N(0,t] = k] = \frac{(\lambda t)^k e^{-\lambda t}}{k!} $$ and which is stationary. Stationarity is a little more involved to go into here, but in this context you can think of it as meaning that the measure of two different intervals of equal length is the same, thus $$ P[N(s,s+t] = k] = \frac{(\lambda t)^k e^{-\lambda t}}{k!}, \ \ \forall s. $$