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Two disjoint unit discs $D_1$ and $D_2$. Inside them there are random poisson points with intensity $\lambda$. For a given real $r>0$, what is the probability that there exist a poisson point $x_1\in D_1$ and a poisson point $x_2\in D_2$ such that $\|x_1-x_2\|_2 \le r$? Couldn't find any existing results on this.

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The number of pairs $(x,y)$ is not a Poisson random variable. –  user49706 1 hour ago
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If $M$ is the $4$-dimensional Lebesgue measure of $\{(x,y) \in D_1 \times D_2: \|x - y\|_2 \le r\}$, then the number of such pairs of points is a Poisson random variable with parameter $M \lambda^2$, so the probability that there is at least one is $1 - \exp(-M \lambda^2)$.

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But how to compute $M$? –  user32828 Apr 7 '13 at 11:58
    
Integrate over $x \in D_1$ the area of the intersection of $D_2$ with the disk of radius $r$ about $x$. I doubt that you'll end up with a closed-form expression. –  Robert Israel Apr 8 '13 at 18:41
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