I am going to frame this question in terms of a pdf and quantile functions for the size distribution of income, because that is where I have currently encountered it, but similar questions in quite different contexts have puzzled me on several occasions in the past.
Suppose I have a pdf f(x) or a CDF F(x) that represents the size distribution of income. Income is a flow of money (dollars or some unit of currency), per some unit of time, to particular individuals. I therefor think the proper unit of measurement is dollars per person-year.
In order for the probability to come out in dimensionless units, f(x) must be measured in person-years per dollar. Integrating this over x also yields a dimensionless share for the CDF because the units of an integral are the product of the units of the integrand f(x), here measured in person-years/dollar, and of the differential dx, here measured in dollars/person-year.
Integrating xf(x) therefore puts the units on the mean income in $/person-year; this seems reasonable.
From this I have drawn conclusions on the units and interpretation of the quantile density function q(y), quantile function Q(y), and the integral of yq(y)dy (i.e. the the inverse function analoge of the partial moment function of order 1) which if correct would be very helpful to me, but in which I have limited confidence. I’d be grateful if someone could confirm or deny these conclusions, perhaps in the context of some discussion/explanation of interpretation of dimensional analysis in the the context of distribution functions.
I believe the units of the y axis and of q(y) are necessarily the same as those in the previous analysis, i.e. the y values are still measured in person-years per dollar and x = q(y) is still measured in $/person-year. This implies that integrals of q(y) over some interval in general, and Q(y) (which is equal to the integral of q(y) from 0 to y) in particular, are both measured in unitless shares. I am somewhat dubious of this and my earlier conclusion, however, because I believe that F(x) is in unitless shares of total income, while Q(y) is in unitless shares of the population for a year. This seems odd to me, however, as I wonder if the fact that these two shares are shares of different things should not be reflected in the units.
Finally, this would have the integral of yq(y)dy in units of person-years per dollar. But I am having trouble interpreting this. The corresponding x quantity is the mean (or for a bounded x, the partial mean) of income per person-year. So this would be the mean number of person-years--for what? for a single dollar? for the mean income?
Consider the truncated version of this function, where the integral of yq(y)dyfrom zero to q* is divided by F(q*). The corresponding truncated function of x yields the mean for those with incomes of x* or less, and $/person-year seems like reasonable units for a mean income. But I want the the y version to come out in number of people, or number of person-years, corresponding to the share of person-years over the same interval, and the "per dollar" in the denominator does not have an interpretation which is obvious to me.
