Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning
$$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$
Now assume $\Phi(f_X)$ is also a probability distribution where $\Phi$ is some function-of-functions of $f_X$. How does one find a $Q$ such that $Q(x)$ is a random variable with distribution $\Phi(f_X)$?
Now in here: Distribution of a function of a random variable
I had learned that if $Q(x)$ is invertible (will consider non-invertible case later) then the distribution of $Q(x)$ is
$$ f_X(Q^{-1}(x))\left| \frac{1}{Q'(Q^{-1}(x))} \right| $$
So to find such a $Q$ we simply have to solve:
$$ f_X(Q^{-1}(x))\left| \frac{1}{Q'(Q^{-1}(x))} \right| = \Phi(f_X)$$
I don't suspect this is easily solvable. But perhaps someone has done research on these types of equations, would anyone be able to point to a source, and then I can descend into my own rabbit hole from there.