I have a doubt on this theorem:
The probability integral transform states that if ${\displaystyle X}$ is a continuous random variable with cumulative distribution function ${\displaystyle F_{X}}$, then the random variable ${\displaystyle Y=F_{X}(X)}$ has a uniform distribution on $[0, 1]$.
The theorem is longer, but I don't understand this first part. If I have a continuous $F(X)$ and I set $Y=F(X)$, then this implies that $Y\sim U(0,1)$? Why?
Thanks!