Suppose we draw a number $x$ uniformly distributed on $(0,1)$, what is then the following distribution. Furthermore, calculate $F(y)$ and $f(y)$.
$$y = \dfrac{x}{1-x}$$
This is a question I came across. Looks very simple, but I just simply do not know. Especially the functions aren't specified anywhere. Do they have a general meaning?
Note that, for every $y\geqslant0$, $$[Y\leqslant y]=[X\leqslant y/(1+y)], $$ hence $$ F(y)=y/(1+y),\qquad f(y)=\mathbf 1_{y\gt0}/(1+y)^2$$