@user14330 says; "The first raw moment of the ratio should be 1." ... It appears that your hypothesis is that if $X$ and $Y$ are iid random variables, then $E[X/Y] = 1$. Unfortunately, your hypothesis is wrong.

The above is all that is needed, other than the requisite software.

Using the mathStatica package for Mma, define the parent pdf as say: f = 1; domain[f] = {x, 0, 1}; . Then, in the $k = 2$ case, the joint pdf of the first 2 order statistics is given by: g = OrderStat[{1, 2}, f], and the desired cdf is simply: Prob[$x_1 + x_2 < z, g$]. For the first 3 order statistics, it is: g = OrderStat[{1, 2, 3}, f] etc

@user64494 We are interested in the joint pdf of $(X_{(1)}, X_{(2)})$ -- not just the isolated pdf of say $X_{(1)}$ on its own, and $X_{(2)}$ on its own. This is because there is dependency between $X_{(1)}$ and $X_{(2)}$ which your model is not capturing.

@user64494 I haven't looked at your Maple work, but your Mathematica working is incorrect: this is because it is incorrect to treat your $x$ as the 1st order statistic and your $y$ as the second order statistic ... you need the JOINT pdf of the 1st and 2nd order statistics. The OP's result seems correct to me, and very neatly stated too.

You may also wish to look up the Loresnz Curve and the Gini Coefficient - both standard measures of statistical dispersion, variously used for modelling income or wealth inequality.

I think this is very confused. I don't agree that you can calculate the cf from some sample means; even if you had the actual theoretical moments, I don't see how you could reconstruct a closed form cf from the moments (without additional information). One cannot recover anything in the manner set out here.

Which people???

More pertinently, why do random variables need us?

[ and mode at 1/2 ]

And the result should follow conceptually ... the $\text{Bates}(n)$ distribution has a constant mean at $\frac12$, and as $n$ increases, the pdf becomes more peaked ... it has variance of $\frac{1}{12n}$, so as $n$ increases, the distribution narrows and converges upon the mean , i.e. if $Z \sim \text{Bates}(n)$, then $P(Z < c)$, for $c > 1/2$ (the mean) must be increasing in $n$.

You want Bates distribution ... not Irwin-Hall.