If I think CDF as a random variable, what is the distribution of CDF ? Is it uniform ?
Let $Y:\Bbb R \to \Bbb R$ and $X: \Bbb R \to \Bbb R$ where the latter is defined by $X(r)=F_Y(r)=P(Y \leq r)$ (so the range of $X$ is $[0,1]$ now). Now if I ask the question about the probability $P(X \leq c)=P(r:X(r)<c)$.
Note that Sample space is $\Bbb R$ here. Does this make sense now ?