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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'?

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Yes both your versions are correct.

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 $g$.

The terminology "change of measure" is rather related to "absolutely continuous measures", which is something completely different, though both create a new measure from a base measure.

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 $E$.

Don't mind my notation, it's very likely you are used to another one.

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