I have looked at how to calculate Inverse Probability Functions from Cumulative Probability Functions, and am familiar with the concept that they are . . . well, inverses. However, I get stuck in actually inverting them.
My question is, given the Inverse Probability Function $e^x$, how can I calculate the Cumulative Probability Function?
For a simple example we can use the symmetry of inverse functions.
$e^p=F^{-1}(p),p\in[0,1]$ and $(F^{-1})^{-1}=F$ then $F(x)=ln(x),x\in[F^{-1}(0),F^{-1}(1)]$
If cdf is not continuous, generalized inverse would not be strictly increasing, $F^{-1}(p)$ being a minimum if there is no $x$ for which $F(x)=p$. The graph on page 8 of https://math.bme.hu/~nandori/Virtual_lab/stat/dist/CDF.pdf demonstrates that we cannot know $F$ if $F^{-1}$ is not strictly increasing.
