If I have two independent random variables $X,Y$ I know that the following holds: $$\text{E}[g(X)h(Y)]=\text{E}[g(X)]\text{E}[h(Y)]$$ for every $g$ and $f$.

Does the same apply for the variance? Meaning: does the following hold too: $$\text{Var}(g(X)h(Y))=\text{Var}(g(X))\text{Var}(h(Y))?$$



No. If $X=1$ and $g(x)=x$ then $Var (g(X))=0$ so RHS is $0$ whereas LHS is $var (h(Y))$.

However the identity is true if $Eg(X)=0$ and $Eh(Y)=0$.

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