# Variance of two indepented R.V.

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: $$\text{Var}(g(X)h(Y))=\text{Var}(g(X))\text{Var}(h(Y))?$$

Cheers

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 $$Eh(Y)=0$$.