I am doing a lot of work with lognormal RVs.
I am trying to get my head around the formal mathematics of the non-linear transform of a random variable, particularly where there isn't any 'wraparound' (as there would be with $X^2$ if $X$ were normally distributed), but simply 'compression' or 'expansion' of the X axis.
The lognormal distribution is an excellent example of that. If $X$ is distributed as $N(\mu, \sigma)$, then $Y=e^X$ is lognormal.
Intuitively it is easy to get the basic idea: if one is thinking about the pdf, roughly speaking, equal-sized increments from the pdf of $X$ when $X>0$ get mapped over a bigger region in 'Y-space'; similarly, when $X<0$, things get compressed. Wikipedia has a great graphic on this idea: https://en.wikipedia.org/wiki/Log-normal_distribution#/media/File:Lognormal_Distribution.svg
I am working with both exact and numerical methods, and am looking for the general abstraction that covers this. For instance, given an (arbitrary) distribution in 'Y-space', I would like to map it backwards into 'X-space'. Intuitively, I would like to take the 'logarithm' of Y if, instead of being lognormal, Y were (say) a Student's t distribution.
I note that such transforms are not trivially obvious. For example, if one wants to go from a lognormal RV $Y$ with known mean and standard deviation, to map back to the normally distributed RV $X$ underlying it, the formulas are not obvious - also, as noted in Wikipedia, they are:
I suspect I am in the area of the Radon-Nikodym derivative, but my knowledge of measure theory is too incomplete to be sure.