Consider a line of the interval $[0,2]$ that gets divided into two parts by randomly (according to $Uniform([0,1])$ choosing one point $w$ of the interval $\Omega := [0,1]$.
Let $X:\Omega \rightarrow \mathbb{R}$ be defined as the quotient $l_1/l_2$ of the shorter segment $l_1$ and the longer segment $l_2$.
First, I defined $X$ in terms of $w \in \Omega$ as $X = \frac {w}{2-w}$.
Is $\mathbb{P}([0,1]) = \frac {1-0}{2-0} = \frac {1}{2}$ correct? What would $\mathbb{P}_X$ be?
I now wonder how to determine the cumulative distribution function $F_X(x)$. Let's say we distinguish between $x < 0$, $0 \leq x \leq 1$ and $x > 1$.
Is the function $X = g(W) = W/(2-W)$ one-to-one for $W \in [0,1]$? Why or why not? If it is, then apply the transformation theorem $$f_X(x) = f_W(g^{-1}(x)) \left|\frac{dg^{-1}}{dx}\right|.$$ If not, compute $$\Pr[X \le x] = \Pr[g(W) \le x] = \Pr[W \le x(2-W)] = \ldots$$