I have a cdf $F(x)$ defined over 0,1. I have a function, $q(x)$, which returns a number between $x$ and $1$. I would like to define a new cdf, $G(x)$, such that $G(q(x)) = F(x)$. I would think there is a simple way to do this, but I can't figure it out. Can anyone help me with the process?
Thank you!