# Finding variance of a random variable given by two uncorrelated random variables

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.

• So you think the variance of $-4X$ is $-4$ times the variance of $X$? Nothing preoccupying there? – Did Feb 6 '19 at 10:49
• I don't see what else it would be? Im new to statistics so Im probably missing something obvious. – Pame Feb 6 '19 at 11:34
• "Im probably missing something obvious" Indeed you are! Recall that every variance is nonnegative. – Did Feb 6 '19 at 18:42

In general, for a random variable $$X$$ and for $$a,b\in\mathbb{R}$$, we have that
$$Var(a+bX) = E[(a+bX) - (a+bE(X))^2] = E[b^2(X-E(X))^2] = b^2Var(X)$$