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I need help to find the finite distribution (fidi) of a homogeneous Poisson point process for overlapping set. For example:

Let $\Phi$ be a uniform Poisson point process of intensity $\lambda$ on $\mathbb{R}$. I want to determine $\mathbb{P}(\Phi(B_1) = n_1, \Phi(B_2) = n_2)$ for $B_1 = [0,2]$ and $B_2 = [1,3]$. Here, $n_1$ and $n_2$ are non-negative integers.

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    $\begingroup$ What if you define the disjoint sets $E,F,G$ by $E=B_1 \cap B_2^c$, $F=B_1\cap B_2$, $G=B_2 \cap B_1^c$? For this example, that is $E=[0,1)$, $F=[1,2]$, $G=(2,3]$. $\endgroup$
    – Michael
    Sep 12, 2016 at 13:36
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    $\begingroup$ What if? Well, then, problem solved. $\endgroup$
    – Did
    Sep 12, 2016 at 13:42

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