Assume we put points on the two dimensional $x-y$ plane according to Poisson distribution, consider $r_1$ is the location of the closest point to origin and $r_2$ is the location of the second closest point to origin. Are $r_1$ and $r_2$ independent? How about for a general distribution (not Poisson)?
To solve this, I am thinking of finding the CDF of $r_1$ and $r_2$ and to see if the following holds:
$\mathbb{P}(r_1<x_1 \,\cap r_2<x_2) = \mathbb{P}(r_1<x_1) \mathbb{P} (r_2<x_2)$
My questions:
- Is this the right path?
- According to Poisson distribution and its special characteristics, can we find the independence faster?
- How about the general case where the distribution is not Poisson?
Thanks