Q:

Finding cumulative distribution function

Lets say I have a Random Variable:
$ \Pi  \sim\mathcal U[\frac{1}{2},1)$
And we also know that
$\Xi =  \ln\left(\dfrac{\Pi}{1-\Pi}\right)$
How can I find the cumulative distribution or the density function of $\Xi$? 
thank you 

A:

If $\Xi = \ln (\Pi/(1-\Pi))$, then $\Pi = 1/(1+e^{-\Xi})$ 
( Further we note that $\Xi$ is a monotonic increasing and invertible function of $\Pi$ over the support interval. ) 
If $\Pi\sim\mathcal U[\tfrac 12, 1)$ then $\mathsf P(\Pi\leq y)= 2y\cdot\mathbf 1_{y\in[1/2;1)}+\mathbf 1_{y\in[1;\infty)}$, and so: $$\mathsf P(\Xi\leq x) ~{= \mathsf P(\Pi\leq 1/(1+e^{-x})) \\ =\lower{1ex}\ldots}$$
Find the pdf by taking the derivative with respect to $x$.

