a) Let $X$ and $Y$ be two uncorrelated random variables. Assume $Var(X) = 1.55$ and $Var(Y) = 0.8$. What is the variance of the random variable $Z = -4X + 5Y - 6$?
b) What if $X$ and $Y$ are correlated with $Cov(X,Y) = 0.6$?
For a) since the variables are uncorrelated I thought you could just sum the variances of the variables? So $$Var(Z) = -4Var(X) + 5Var(Y)$$ However this produces an incorrect answer.
Watch out! While it is true that the variance of a sum of uncorrelated random variables is indeed the sum of their variances, as Did suggested you must give special attention to the constants scaling the random variables.
In general, for a random variable $X$ and for $a,b\in\mathbb{R}$, we have that
$$Var(a+bX) = E[(a+bX) - (a+bE(X))^2] = E[b^2(X-E(X))^2] = b^2Var(X)$$
which means that variance is scaled quadratically wrt its random variable's multiplicative constant, and not at all with respect to additive constants. Can you figure it out from here?