Let $X_{(1)}\leq X_{(2)}\leq \cdots X_{(n)}$ be the order statistics for a random sample from a continuous distribution with c.d.f. $F(x)$ and density $f(x)$. Define $U_{i}$, $i=1,2\ldots,n,$ by $$U_{i}=\frac{F(X_{(i)})}{F(X_{(i+1)})}, \quad i=1,\ldots,n-1, \: \mbox{and }\: U_{n}=F(X_{(n)}).$$

Find the joint distribution of $U_{1},\ldots,U_{n}$.

**Remark:** I need a suggestion or someone tell me in which book I can find this exercise or related theory that allows me to solve it.