So I'm trying to show that ${rm Var}(Z) le 2({rm Var}(X)+{rm Var}(Y))$ for $Z = X + Y$. This seems to be pretty easy to show given that $X$ and $Y$ are uncorrelated. But I'm running into trouble at this step:
$$
{rm Var}(Z) = {rm Var}(X) + {rm Var}(Y) + 2E[XY] – 2E[X]E[Y]
$$
Normally you could say, $X$, $Y$ uncorrelated $rightarrow E[XY] = E[X]E[Y]$, but when you cannot do this, I'm lost. Any tips?
Best Answer
I don't want to write out the full answer because this looks a lot like homework or self-study (and if it is indeed homework or self-study, please add the homework or self-study tag).
Hint: the maximum value that $operatorname{cov}(X,Y)$ can have is $sqrt{operatorname{var}(X)cdotoperatorname{var}(Y)}$ (the minimum value is $-sqrt{operatorname{var}(X)cdotoperatorname{var}(Y)}$). Use this together with $$operatorname{var}(Xpm Y) = operatorname{var}(X)+operatorname{var}(Y) pm 2operatorname{cov}(X,Y)$$ to see if you can get anywhere with this exercise.
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