X and Y are two random variables, what is the standard deviation of X - 2Y

Question

I have:

E(X) = 10
E(Y) = 12
Var(X) = 4
Var(Y) = 9
covariance = 2

I know that for:

Var(X + Y) = Var(X) + Var(Y) + 2covariance

Which would be useful if I could do this:

Var(X - 2Y) = Var(X) + Var(-2Y) + 2*covar(X, -2Y)

But I do not have the variance of -2Y nor the covariance covar(X, -2Y).

Answer 1

The formula you have for the variance of $X+Y$ generalizes to the variance of the linear combination $\alpha X+\beta Y$. We have in general $$\text{Var}(\alpha X+\beta Y)=\alpha^2 \text{Var}(X)+\beta^2\text{Var}(Y) +2\alpha\beta\text{Cov}(X,Y).$$ The proof is essentially the same.