The Pareto distribution can be used to give a pdf for the wealth of a person chosen randomly from a population. (In fact, this was its origin. See, for instance, http://en.wikipedia.org/wiki/Pareto_principle ).
I would like to explore the reciprocal question: Given the total amount of wealth in a population, what is the pdf of the portion that a randomly chosen person has. I conjecture that this is simply a constant times the Pareto distribution.
More interestingly: What is the shape of the distribution curve, if the richest person would be at the 0 point on the x axis, the next richest person to the right, and so on - we would see a monotonically decreasing curve. But what is its shape? What is its derivative?
It's quite likely that I'm not phrasing that question properly. Let me ask a more basic question: What is the appropriate terminology to explore the question? Give a probability distribution applied many times over, what is the shape of the resultant allocation curve?