Does this distribution relation have a name: $\mathbb E(X^n)=b^n \mathbb E(Z^n)$

Question

More precisely, if $X$ and $Z$ are distributions so that $X=bZ$, then $\mathbb E(X^n)=b^n \mathbb E(Z^n)$. I found it on this site and would like to know "where it comes from".

Answer 1

In general, if $Z$ is a random variable in the probabilistic space $(\Omega,\mathcal{F},P)$, and $X=bZ$then, $$\int_\Omega X^ndP=\int_\Omega b^nZ^ndP=b^n\int_{\Omega}Z^ndP$$.

Thus $E[X^n]=b^nE[Z^n]$.