I have a question concerning a rather specific posterior. It should be a simple application of Bayes' Theorem. However, I am always confused here.
I try my best to describe the situation.
There are n draws from a continuous distribution F over some support. Then the following algorithm starts that divides the values into three different urns.
All values below some threshold x1 are put into the first urn. All values between x1 and x2 are put into a second urn. All values above x2 are randomly put into either the second or a third urn with probability 1/2. Then, only the highest value remains in the second urn, all others are put into the first urn. The same happens with urn three.
Now the question is: If I only know the distribution F, the number n, and the design of the algorithm (including the thresholds), what is the distribution of a value randomly drawn from the first urn conditional on A) urns 2+3 are empty, B) urn 2 is non-empty, C) urn 3 is non-empty?
A) seems to be obvious: Just F(x)/F(x1), i.e. just the prior distribution conditional on all values being below x1.
B) and C) seem to be a more advanced Bayesian posterior.
I appreciate your help :3