How do we understand $F_Y(Y)$? In probability, we know that $F_Y(y) := P(Y\le y)$ is is the probability that the random variable $Y$ is less than or equal to a given real value $y$. But how do we understand the notation $F_Y(Y)$. For example. If Y follows a Gaussian distribution, what is its $F_Y(Y)$?
 A: If $Y$ is a random variable and $G$ is a fixed (i.e. not random) function, then $G(Y)$ is itself a random variable.
For the standard normal (or "Gaussian") distribution, you have
$$
F_Y(y) = \Pr(Y\le y) = \int_{-\infty}^y \frac 1 {\sqrt{2\pi}} e^{-u^2/2} \, du
$$
So $F_Y(y)$ is the value of the cumulative probability distribution function at a particular number $y.$
Now look at $F_Y(Y),$ which is itself a random variable. Observe that $F$ is continuous and strictly increasing, and approaches $0$ at $-\infty$ and approaches $1$ at $+\infty.$ So for $0<x<1,$ you have
$$
\Pr(F_Y(Y) \le x) = \Pr(Y \le F_Y^{-1}(x)) = F_Y(F_Y^{-1}(x)) = x.
$$
For $x\ge 1$ you have $\Pr(F_Y(Y)\le x) = 1$ since all values of $F_Y$ are less than $1,$ and for $x\le0$ you have $\Pr(F_Y(Y)\le x) = 0$ since all values of $F_Y$ are more than $0.$
So you have
$$
F_{F_Y(Y)}(x) = \begin{cases} 0 & \text{if } x \le 0, \\ x & \text{if } 0<x<1, \\ 1 & \text{if } x\ge1. \end{cases}
$$
In other words $F_Y(Y)$ is a random variable that is uniformly distributed on the interval $[0,1].$
