# Why is the co-variance multiplied by two when summing the variances of two correlated random variables?

As I understand, when trying to sum the variances of two correlated random variables, the formula is such:

$Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y)$

My question is, why is the last term multiplied by two? I feel like I am just missing something really obvious, but I can't wrap my head around why it's necessary. Maybe someone can explain for me?

• I remember a proof of this using explicit $x_i's$ and $y_i's$ it was long but worth it, just try. – Mann Jun 2 '15 at 6:59

## 1 Answer

We have that $$\operatorname{Var}(X+Y) =\operatorname E(X+Y-\operatorname E(X+Y))^2 =\operatorname E(X-\operatorname EX+Y-\operatorname EY)^2$$ and $$(a+b)^2=a^2+2ab+b^2$$ for $a,b\in\mathbb R$.