If X have a uniform dist. on the interval (0,1)
How do you find cdf and pdf of $$Y = \frac{X}{1-X}$$
I know that pdf is
$$f(x) = \left\{ \begin{array}{cl} 1 & 0<x<1 \\ 0 & o.w \end{array} \right.$$
I do not know how to interpret $\ \ Y = \frac{X}{1-X}$ and I know my interval changes when I use $$\ \ Y = \frac{X}{1-X}$$ but it seems like when I plug $1$ it becomes undefined.
Hint: Note that $$ P(Y \le y) = P \left( \frac{X}{1-X} \le y \right) = P \left( X \le y(1-X) \right) = P(X(1+y) \le y ) = P\left( X \le \frac{y}{1+y} \right) $$ Can you continue from here?