Sum of variances of 2 correlated random variables

Question

I have 2 random variables (RVs) $X$ and $Y$ which are correlated by $\rho$.

$$X, Y \sim N(0, \sigma_1)$$.

X and Y are both multiplied by $Z$ which is also a normal RV, independent from $X$ and $Y$ ($Z \sim N(0, \sigma_2) $).

$$ Out = Z \cdot X + Z \cdot Y $$

If I isolate the variance of Z, I get the sum of of the variance of X and Y.

\begin{align*} var[Out] &= var[Z]var[X] + var[Z]var[Y] \\ &= var[Z] \cdot \left( var[X] + var[Y] \right) \end{align*}

Since X and Y are correlated, does this become ?

\begin{gather*} var[Out]= var[Z] \cdot \left( var[X] + var[Y] + 2cov[X,Y] \right) \end{gather*}

Or do I just leave $var[X] + var[Y]$ as is ?

Answer 1

Since $X$ and $Y$ might not be independent, $ZX$ need not be independent from $ZY$. So your first computation needs a $2\operatorname{cov}(ZX, ZY) = 2\operatorname{var}(Z) \operatorname{cov}(X, Y)$ term, where the equality holds because $Z$ is independent of both $X$ and $Y$. Now if you factor out $\operatorname{var}(Z)$ you get the same expression as the second method.