Let XXX be continuous with quantile function Q(p)=ln(p1−p)Q(p) = \ln\left(\frac{p}{1-p}\right)Q(p)=ln(1−pp) for 0<p<10 < p < 10<p<1. What is the PDF f(x)f(x)f(x)?
f(x)=1(1+ex)2f(x) = \frac{1}{(1+e^x)^2}f(x)=(1+ex)21
f(x)=ex(1+ex)2f(x) = \frac{e^x}{(1+e^x)^2}f(x)=(1+ex)2ex
f(x)=11+e−xf(x) = \frac{1}{1+e^{-x}}f(x)=1+e−x1
f(x)=e−x(1+e−x)−2f(x) = e^{-x}(1+e^{-x})^{-2}f(x)=e−x(1+e−x)−2