# Moments of derived distribution

Given r.v. X with PDF $$f_X(x)$$ how can I find the moments of PDF of derived r.v. $$Y = g(X)$$ where $$g(x)$$ is continuous non-monotonic function?

I suspect that all I need to do is to calculate $$M_n = \int\limits_{-\infty}^{+\infty} g^n(x) \: f_X(x) \: dx$$

If it is true then it looks much better alternative to an attempt to find $$f_Y(y)$$ for r.v. Y itself and then use n-th moment definition:

\begin{align} M_n = \int\limits_{-\infty}^{+\infty} y^n \: f_Y(y) \: dy \end{align}

Additional question on terminology: can transition from $$f_X$$ to $$f_Y$$ be called 'change of measure'?

No I wouldn't call that change of measure but rather "change of variables",or say that $$f_Y(y)dy$$ is the pushforward measure of $$f_X(x)dx$$ by the map $$g$$.
With pushforward measure you have a base measure $$\mu$$ on $$E$$, and a map $$f : E\to F$$ and you obtain a measure $$\mu(f\in dy)$$ on $$F$$.
With absolutely continuous measure you have a base measure $$\mu$$ on $$E$$, and a map $$f : E\to \Bbb R_+$$ and you obtain a new measure $$f(x)\mu(dx)$$ on $$E$$.