Dr. Memnus Lagbane Soultooth the Mad

I'd like your thoughts on a problem. (Here, all "randomly chosen" numbers are taken from the same continuous real distribution.) Suppose you have a set S of n randomly-chosen numbers, and a set D, initially empty. Begin by removing the largest number from S and placing it in D, and adding another random number to S. Now take the largest number from S that is smaller than the last number you chose, place it in D, and replace it with a random number. Repeat until you've taken the smallest number from S.

- What is the distribution of numbers in D?
- What is the expected value for the size of D?
- Bonus question (I know the answer to this one!): Why is Brian thinking about this problem?

I'm about to start in on a Monte Carlo estimator of the problem, but I'd like some theoretical input.

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