If I have a function $f(x)$ with a domain $[a,b]$ and range $[c,d]$, how do I find the moments of the values the function attains in the domain?
More importantly, how do I find the PDF for the function's values?
I'm motivated by a problem Functional Integration. I can find both the mean value of a functional $S[f]$ and the standard deviation. However, I'd like to find the PDF for the values the functional attains.
For $f(x)=x$ on $[a,b]$, I can make an educated guess and state the PDF is $\rho(f(x))=\cfrac{1}{b-a}$.
But I don't really know what the PDF looks like for even moderately more complex examples like $f(x)=x^2$ or $f(x)=\cos(x)$.
(I'll take references/links; but an example is always appreciated)