A proposition in my textbook goes:
Let $Z=F(X)$; then $Z$ has a uniform distribution on $[0,1]$.
The proof is as such:
$$P(Z \leq z) = P(F(X) \leq z)=P(X \leq F^{-1}(z)) = F(F^{-1}(z))=z$$ which is a uniform CDF.
This proposition has left me very puzzled and unable to comprehend further. Can any kind and intelligent soul explain to me what this proposition means in plain, intuitive terms? I have checked other threads on this, but am unable to comprehend it from a basic level. Any help is much appreciated! Thank you!
Let $U \sim Uni(0, 1) \implies F_U(u) = \mathbb{P}(U < u) = u ~ \mathbb{I}_{\{u \in (0,1)\}}$, where $\mathbb{I}$ is the indicator function.
For any random variable $X$ with c.d.f $\mathbb{P}(X<x) = F_X(x)$
$$\mathbb{P}(U < F_X(x)) = F_X(x) = \mathbb{P}(X < x) = \mathbb{P}(F_X(X) < F_X(x))$$ Now let $y = F_X(x)$, then $$\mathbb{P}(U < y) = \mathbb{P}(F_X(X) < y)$$, so for any random variable $F_X(X)$ has a $Uni(0, 1)$ distibution.