Say I have a bag of beans, with different lengths, but following a normal distribution with some mean and some standard deviation.
If I were to take out the beans in pairs, and cut the longer one in half, leave the shorter one alone, and put them both in a different pot (so they won't be subject to more than one inspection or cut), and continue doing this for all the beans, how will the distribution change? (An acceptable result is finding the new mean and standard deviation as functions of the old ones - a simulation is interesting but I can do that myself, and wouldn't give that satisfaction of understanding that a real solution would bring).
Context: I thought of this problem a number of years ago while doing a first year undergraduate course in statistics, while chopping beans for dinner. I later asked my tutor, but he didn't know how to solve it (or didn't have time). I thought of it again a few days ago, again while chopping beans. I never did any further studies in statistics and have forgotten 90% of what I learned so I'm not really equipped to begin solving it myself. Hoping I can get closure on this topic :-)