# Understanding the Pareto distribution as applied to wealth

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?

• There is a problem with your first sentence, which probably is the cause of the further problems. Is this your sentence, or is it citation? What is the pdf of wealth for one person? Pdf combines information about many values random variable takes. Information about one person's many values wealth is usually private and generally only available to tax offices. Usually given the income distribution of whole population and the wealth of particular person you can say what is the probability of finding wealthier or poorer persons. Is this what you are saying? – mpiktas Feb 28 '11 at 9:04
• The conjecture in the second paragraph is false. The portion of wealth is a number, distribution times the constant is function. – mpiktas Feb 28 '11 at 9:07
• Thanks, mpiktas - I believe I fixed them. My basic question is: "Help me understand the relationship between P(X has this much wealth) distribution and distribution of wealth in the society as a whole. – user3463 Feb 28 '11 at 9:28
• @Marcus, I find your usage of term distribution a bit confusing. P(X has this much wealth) is a number, $P(X<x)$ for given $x$ is a distribution of $X$. Do you per chance need something like Lorenz curve? – mpiktas Feb 28 '11 at 9:58
• The Lorenz curve is very appropriate. But it's cumulative. I think I'm looking for the derivative of the Lorenz curve (of a Pareto distribution or in general). What is that? – user3463 Feb 28 '11 at 14:57