# why double Sigma summation?

The lecture slides is about covariance. One page it is

on the other page, it is

Why there are double Sigma summation on the first one?Is it just a typo?

• It is simply notation. There is a double summation. One for all the $x$ and one for all the $y$.
– Eff
Commented Jan 7, 2016 at 13:50
• It is a typo. One can either use one summation sign indexed by $(x,y)$ or two summation signs indexed by $x$ and by $y$ respectively, but mixing these two options is absurd.
– Did
Commented Jan 7, 2016 at 13:52

I think a very clear definition of covariance is:

$Cov(X,Y) = \sum\limits_{x \in X}\sum\limits_{y \in Y} (x- \mu_X)(y-\mu_Y)f(x,y)$

If you define $S = X \times Y = \{(x,y) | x \in X, y \in Y\}$ (cartesian product).

Then you can rewrite the definition as:

$Cov(X,Y) = \sum\limits_{(x,y) \in S} (x- \mu_X)(y-\mu_Y)f(x,y)$

I think the first equation in your post is a confusion between the two definitions written in this answer. It is not even clear to me what summation sign $(x,y) \in S$ belongs to. However, a lot of notation abuse like this is made, if it is clear what is being meant. Which was obviously not the case here since you were confused.

It is a double summation, one over $x$ and one over $y$, so strictly speaking you need two $\Sigma$ signs on both slides.

Sometimes people leave only one just for easier notation.

• "strictly speaking you need two Σ signs" No, see comment.
– Did
Commented Jan 7, 2016 at 13:53
• $(x,y) \in S$ is only one element. So I would argue only one summation sign is needed. A summation over all $(x,y) \in S$. Commented Jan 7, 2016 at 13:55