# Intuition (geometric or other) for $Var(X+Y) = Var(X)+Var(Y) + 2 \ Cov(X, Y)$

Is there any way to make sense out of this formula intuitively?

I rederived it algebraically (took me a while...), which made me happy because I used to be incapable of doing that kind of stuff, but I still have no intuition for it.

Geometric intuition would be best, but anything is very welcome.

• Here's a trivial example that may or may not help with intuition: $Var(X + X) = Var(X) + Var(X) + 2Cov(X, X)=Var(X) + Var(X) + 2Var(X)=4Var(X)=Var(2X)$. – lmo Jun 27 '16 at 16:19
• Geometric intuition is straightforward in vector representation of (centered) variables in subject space. There, vector length is st. deviation and angle b/w vectors is Pearson $r$ (see). Then, it appears that the st. dev. of the sum X+Y is the longer axis of the parallelogram, and by its trigonometric law you get the formula you ask about. See the 1st pic with formulas here. – ttnphns Jun 27 '16 at 16:40
• See also a not the same, but a related question. It is about correlation between X and X+Y (when X and Y are uncorrelated, as a particular case). – ttnphns Jun 27 '16 at 16:58
• As for algebraic way, the hint is $(a+b)^2=a^2+b^2+2ab$, from school. One can start from here. – ttnphns Jun 27 '16 at 20:29
• @ttnphns, thank you for the links. I remember just enough Linear Algebra to make sense of them. – jeremy radcliff Jun 28 '16 at 8:05

1. More mathematical way

The variance of two variables, X and Y, can explained by separating it into four sections: how X varies independently, how Y varies independently, how X varies due to changes in Y, and how Y varies due to changes in X. Essentially, all changes in X and Y have to be caused by one of these four things.

That gives us:

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

which can be simplified, because $Cov(X,Y) = Cov(Y,X)$ into:

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

1. Not mathematical way

This is a vast oversimplification and is not mathematically correct, but it helped me with remembering.

Consider the much simpler algebraic equation:

$(X+Y)^2=X^2+Y^2+2XY$

Also consider that:

$Var(X)=\sigma^2(X)$

So:

$[\sigma(X+Y)]^2=[\sigma(X)]^2+[\sigma(Y)]^2+2[\sigma(X)\sigma(Y)]$

• Thank you, that's very helpful, especially the first part – jeremy radcliff Jun 28 '16 at 8:04