I have some unclear questions as follows:
Problem 1:
$ X$: is defined as a random variable, so
$F_{X}(x)$: is its CDF. So what is is the difference between $F_{X}(x)$ vs $F(x)$ ?
Problem 2:
Let $U$~$\mathbf{U}[-1,1]$ (U: uniformly distributed variable) and
let $Z=F_{X}^{-1}(U)$, then $U=F_{X}(Z)$ it is strange if z is a variable of the CDF function.
so what is the meaning of $U=F_{X}(Z)$? (usually, we know $U=F_{X}(x)$ as a CDF of X)
Problem 3:
What is the difference between these definitions? $F_{X}(x)$ vs $F_{X}(X)$