Probability Integral Transform

Let X have a continious c.d.f $F$, and let $Y = F(X)$.
(This transformation from $X$ to $Y$ is called the probality integral transformation)
The distribution of $Y$ is the uniform distribution on the interval [0, 1]

Let $U$ have the uniform distribution on interval [0, 1], and let $F$ be a continuous c.d.f with quantile <