Chapter 5.1, Problem 8E

Let X have a logistic distribution with pdf $f\left(x\right)=\frac{e^{-x}}{{\left(1+e^{-x}\right)}^2},-\infty <x<\infty$.

Show that $Y=\frac{1}{1+e^{-x}}$ has a U(0, 1) distribution. Hint: Find $G\left(y\right)=P\left(Y\le y\right)=P\left(\frac{1}{\left(1+e^{-X}\right)}\le y\right)$ when $0<y<1$.