Let $X$ be a continuous r.v. with p.d.f. $f$ and c.d.f. $F$. Suppose that $f$ is continuous and that $f(x)>0 \, \forall \, x\in \mathbb{R} $. Compute the p.d.f of the r.v. $F(X)$.
I tried the usual method. Let $Y = F(X)$. $F_Y (y) = P(\int_{-\infty}^{X} f(t) dt \leq y)$. But unlike usual, I don't know how to write this as $P(X\leq g(y))$ for some function $g$. The fundamental theorem of calculus is probably not useful.
Edit: The usual method is as follows: Given a r.v. $Y$, we find the c.d.f. $F_Y$ of $Y$. $\frac{dF_Y}{dy}$ is the p.d.f. of $Y$
Please give a hint. Thanks