Let $X$ be a continuos random variable taking values in $[a,b]$ with c.d.f $F_X$ which is strictly increasing on $[a,b]
(a) Show that the random variable $F_X(X)$ has a uniform distribution on $[0,1]$.
(b) Let $U$ be a uniform random variable on $[0,1]$. What is the distribution of the random variable ${F_X}^{-1}(U)$ where ${F_X}^{-1}$ is the inverse of $F_X$?
It has been shown how to solve (a) in the answer to the question in the link below. However, I still don't know how to approach (b).