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According to the definition of Poisson Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed in a restricted area) similar to PPP (An infinite number of nodes distributed in an infinite area)?

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  • $\begingroup$ You have typo in the tile: simlar -> similar. And Possion -> Poisson. $\endgroup$ Sep 13, 2013 at 7:17
  • $\begingroup$ I'm sorry, but thank you for pointing out my mistakes. $\endgroup$
    – xzhh
    Sep 13, 2013 at 9:52

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You can certainly restrict a Poisson point process to a finite region. In a finite area, the number of points will be almost surely finite. If you want to condition on the number of points being $n$, you just get $n$ independent points uniformly distributed over the region.

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  • $\begingroup$ If I restrict a Poisson point process to a finite region and take a snapshot of the region(assuming there are $n$ nodes in this region at the moment), how are these nodes distributed in the region? Are they independently uniformly distributed ? $\endgroup$
    – xzhh
    Sep 13, 2013 at 8:48
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    $\begingroup$ That's what I said. $\endgroup$ Sep 13, 2013 at 15:05
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A more useful definition of a Poisson point field is:

Let $\mu$ be a (finite or infinite) measure on a space $E$ with a sigma-algebra $\mathcal{B}$. To each set $A\in\mathcal{B}$ with $\mu(A)<\infty$, one assigns a r.v. $X(A)$ counting the number of Poissonian points in it. That random variable is supposed to have Poisson distribution with mean $\mu(A)$, and if $A_1,\ldots, A_n$ are mutually disjoint, then $X(A_1),\ldots,X(A_n)$ are supposed to be mutually independent.

The advantage of this definition is its generality. Also, you can clearly see how this definition is trivially preserved under constraining the space to subsets of $E$.

What you are talking about is the specific case case where the driving measure $\mu$ is Lebesgue or a multiple thereof.

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