correlation of transformed uncorrelated random variables

Question

Let $X$ and $Y$ be uncorrelated random variables and $h$ measurable function. It is true that $EXEY=EXY$, but i dont know whether this statement is true: $Eh(Y)X=Eh(Y)EX$. For independent variables it is true, but what about uncorrelated? Maybe the question should be: if X and Y are uncorrelated, does it imply that $X$ and $h(Y)$ are uncorrelated?

Answer 1

The answer is "no, in general". Say, for $Y$ standard Gaussian and $X=Y^2$ $$ \mathbb E(XY)=\mathbb E(Y^3)=0=\mathbb EX\mathbb EY, $$ but for $h(Y)=Y^2$ $$ \mathbb E(Xh(Y))=\mathbb E(Y^4)=3\neq \mathbb EX\mathbb Eh(Y)=(\mathbb EY^2)^2=1. $$