Let $X$ be a random variable with continuous distribution of $F$. Find the distribution function for the random variable $F(X)$.
This is an exercise from my probability theory with measure theory book but I am not so sure how to approach it. I tried doing things like $P(F(X) \leq x)$, and I got nowhere. I would greatly appreciate some help with this exercise. I am trying to get better at these problems but do not have much help on it.
I have been stuck in this problem for quite some time now so I would really appreciate your support
Since $F$ is increasing $\{t:F(t) \leq x\}$ is a interval of the type $(-\infty, y)$ or an interval of the type $(-\infty, y]$. Note that $P(F(X) \leq x)=P( X \in \{t:F(t) \leq x\})$, so $P(F(X)\leq x)=F(y)$. [ In view of continuity of $F$ the $P(X<y)$ and $P(X \leq y$ are the same]. Now $y=\sup \{t: F(t) \leq x\}$ and it follows by continuity that $F(y)=x$. Hence $P(F(X) \leq x)=x$ for $0\leq x \leq 1$. In other words $F(X)$ has uniform distribution on $(0,1)$.