Understanding notation in proof of probability integral transform

Question

I'm attempting to understand one line of the proof for the probability integral transform as found on wikipedia:

Suppose that a random variable $X$ has a continuous distribution for which the cdf is $F_X$. Then the random variable $Y = F_X(X)$ has a standard uniform distribution.

Proof:

$ F_Y(y) = P(Y \leq y) = P(F_X(X) \leq y) = P(X \leq F^{-1}_X(y)) = F_X(F^{-1}_X(y)) = y $

What I do not understand is the definition of the random variable $Y$, namely why is there a capital $X$ in parentheses, $F_X(X)$, instead of lower-case, $F_X(x)$. More importantly, what does this mean?

I have looked at this post already, and my updated understanding is that $F_X(X)$ represents the distribution of the probabilities of $X$, not the variable itself. So, I believe that $Y$ is the distribution of probabilities of $X$. Is this correct? Or, if not, can someone explain what this difference in notation means?

Answer 1

Because, take $F_X(x)=G(x)$ as a function and we apply this transformation over the random variable $X$, to obtain $Y$. So, in general, if the input is a RV, the output is a RV, i.e. $Y=G(X)$, not $Y=G(x)$. You could say $y=G(x)$, for a specific pair of $(x,y)$ by the way. Therefore, the notation confusion you have doesn't have anything to do with the meaning of $F_X$.