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!