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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!

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    Thanks. In my case, both F and q are monotonically increasing. This helps! I would "accept" if in answer form :)2011-09-10

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(This answer is a copy of points made in comments by @Sasha and me.)

If $F$ is increasing, $q$ must be increasing. Then $G(x)=F(q^{-1}(x))$, where $y=q^{-1}(x)$ is the unique solution of the equation $x=q(y)$. If $F$ is only nondecreasing, $q$ may be only nondecreasing as well, provided $q$ is constant on intervals related to those where $F$ is constant.