Timeline for Probability that uniformly random points in a rectangle have Euclidean distance less than a given threshold
Current License: CC BY-SA 3.0
12 events
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| Feb 10, 2012 at 17:38 | comment | added | zhouzhuojie | @whuber I also found a paper which discussed the expectation of the distance between two points. But I just don't know how to transform it to exactly what I amended in the new description. | |
| Feb 10, 2012 at 17:35 | comment | added | cardinal | @rexzh0u: Ok. Sorry if my recollections were off the mark. I will try to post a complete answer later today or tomorrow. I tend to be painfully slow at that. | |
| Feb 10, 2012 at 17:33 | comment | added | zhouzhuojie | @cardinal Thanks. Actually I went through all the questions on Math.SE, but I still could not find some close to this problem. | |
| Feb 10, 2012 at 17:28 | comment | added | zhouzhuojie | @cape1232 Thanks so much. I modified the question to make it more clear. | |
| Feb 9, 2012 at 22:07 | comment | added | cape1232 | The OP said we have n points. And he wants to know the probability any two of those are within distance d. That sure seems like you can just loop over them. I do not think he asked the more general one: given two points randomly sampled from said rectangle, what is the probability they are closer than d. You have to define randomly sampled to solve this, and then it's similar to Betrand's paradox. | |
| Feb 9, 2012 at 21:18 | comment | added | whuber♦ | The boundaries of the domain (here a rectangle) don't enjoy any translation invariance, so I'm not sure how that's going to apply. (The torus is a different matter.) | |
| Feb 9, 2012 at 20:52 | comment | added | cardinal | @whuber: I believe I have a fairly simple solution for this, but let me check it more carefully before I post it. :) | |
| Feb 9, 2012 at 20:39 | comment | added | cardinal | @whuber: A solution? No. But, I'm almost positive this question appears. :) I'll see if I can find it. At any rate, I'm not sure this problem is so difficult, even in this case. I believe you can use translation invariance to simplify it somewhat. But, I haven't worked out the details. | |
| Feb 9, 2012 at 20:00 | comment | added | whuber♦ | Are you sure there's a solution on math.SE, cardinal? This is a difficult problem due to the edge effects. Maybe there's a solution on the flat torus. | |
| Feb 9, 2012 at 19:40 | comment | added | cardinal | I think you may have misunderstood the OP's question. Also, the desired distribution is unambiguously defined in the question. My comment to the OP hints that there is already a solution on the SE network to this question, hence this one can most likely be closed. :) | |
| Feb 9, 2012 at 19:39 | comment | added | whuber♦ | The question asks about the distribution for iid uniformly distributed points: these are random variables, not any "fixed known pattern," and one cannot just loop over pairs of them! | |
| Feb 9, 2012 at 18:49 | history | answered | cape1232 | CC BY-SA 3.0 |