If, $F^{-1}(u)=\inf\{x\in\mathbb R:F(x)>u\}$ and $U$ is uniformly distributed in $[0,1]$, what is the law of $F^{-1}(U)$ ? ($F$ is a distribution function of some random variable $X$)
How can I determine its distribution function, I don't think that;
$P(F^{-1}(U)\le t)=P(U\le F(t))$
The exercise must have something special,
$\inf\{x\in\mathbb R:F(x)>u\}\le t\Longrightarrow u< F(t)$
How can I show that, there's no gap between $u$ and $F(t)$ $($that $u$ takes all values smaller than $F(t))$