# How can I get the covariance just given the variance?

This is a fairly straight forward problem but I don't know how to calculate the covariance given just the variance of X and Y.

Suppose X and Y are independent random variables such that Var(X)=1 and Var(Y)=2. What is the variance of X+2Y-3?

So far, I have

(1^2)var(x) + (2^2)var(y) + 2(1)(2)cov(x, y)

Which comes out to

1 + 8 + 4cov(x, y)

How can I calculate 4cov(x, y)?

• Hint: If $X$ and $Y$ are independent, then $\mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$. In particular, $\mathrm{Cov}(X,Y) = 0$. – snar Mar 10 '15 at 18:16
• Thank you! Wow, that was very simple. – Abbas Dharamsey Mar 10 '15 at 18:29
• If $X$ and $Y$ are independent their covariance is $0$. – André Nicolas Mar 10 '15 at 18:40
• Correct me if I'm wrong, but I think you just want $V(X + 2Y - 3)$ and you're imagining you need the covariance, which you don't. By independence, $$V(X + 2Y - 3) = V(X) + V(2Y) = V(X) + 4V(Y).$$ – BruceET Mar 13 '15 at 3:01