The probability of two random points on a plane being closer than a certain distance?

Hi,

I've been struggling with the following problem for some time now, and it seems that I need some pro help:

I have a process that generates a point on a plane, and I need to find the pdf of that point. The process goes like this: First, points and are independently generated on the plane with pdfs and . Then, only if the two points are closer than a certain distance , i.e. , I take as my point .

This point is used for Monte Carlo integration, and if I use the pdf it all works nicely. However, I'm having a hard time justifying this particular pdf. I see two possible ways: (1) this pdf is a decent approximation of the real pdf; (2) this pdf is an unbiased estimate of the real pdf. More specifically, is the result of a one-sample Monte Carlo estimation of the integral that appears due to the distance probability.

I'm having a hard time mathematically formulating this pdf. Any ideas? Thanks very much in advance!

Re: The probability of two random points on a plane being closer than a certain dista

Hi,

As far as I understand, you define to be conditioned to be at distance smaller than to , right? In other words, you sample and repeatedly until they are closer than from each other and then you set . Is that indeed it?

If so, first the law of should not symmetric with respect to and as they have different roles. And second there is no meaning to writing as a function of which are not defined in terms of ? So something has to be wrong; it may be my understanding, but let's carry on.

Given my rewriting of the problem, the pdf of is derived as follows (I use a slightly sloppy notation "dx" that stands for an "infinitesimal subset around x"; you can turn it to fully rigorous by writing a Borel set A instead of dx, but I feel this writing with dx gives more feeling of what happens):

hence

so the pdf is the function on the right hand side.

If is continuous at , the integral at the numerator is close to when is small. Then, for small , one may say that the pdf of is close to , where (so that integrates to 1).

I hope this answers your question. Feel free to ask for further details. Or to make your question more specific, if I got it wrong.

Re: The probability of two random points on a plane being closer than a certain dista

Thanks for the reply, Laurent!

I kind of understand what you mean. I'm not sure this is exactly what I want, but it's close, I feel.

In the following, please forgive any incorrect terminology/notations used. I'm really a computer scientist with a shallow background in probability theory.

The generated point will be used for Monte Carlo integration of a function that is defined everywhere where can be (e.g. the whole plane). Actually, I can already use for the MC integration with PDF . Just that I want to have another sampling strategy that takes only if a random point with PDF lands within a distance of .

Therefore, I'm not sure about whether this normalization is needed. Otherwise the nominator looks OK: If I assume that is constant within the -neighborhood around , then I can take out of the integral as and then I get my result .

How does that sound?