# Having trouble understaning the notation co-variances

• Rabolisk

#### Rabolisk

Can someone explain to me how
Cov(x,y)=[X-E(x))(Y-E(y)]

and this would equal 0 under the normal OLS assumption.

I know how to calculate the covariance, but X here is a matrix. So I don't understand the logic of this formula...

Welcome to PF, Rabolisk!

Did you already look at the wiki page?
http://en.wikipedia.org/wiki/Covariance

Your formula does not seem to be quite right.
X and Y could be random variables, but in that case an extra E should be present in your formula (see wiki).

If your X and Y are matrices then your formula should have an extra 1/N in it (see wiki).

There is an interesting analogy between lengths-of-vectors and the variances-of-random-variables.

For vectors X and Y, we have
$| X + Y |^2 = |X|^2 + |Y|^2 - 2|X||Y| \cos \theta$ where $\theta$ is the angle between the vectors.

For random variables X and Y we have

variance(X+Y) = variance(X) + variance(Y) + 2 Covariance(X,Y)

The analogy is even better if we express it in standard deviations:

$( std. dev(X+Y))^2 = (std. dev(X))^2 + (std. dev(Y))^2 + 2 Covariance(X,Y)$

The covariance is roughly analagous to the cosine term. The quantity that would be analagous to $cos(\theta)$ is $\frac{- Covariance(X,Y) }{(std. dev(X)) (std. dev(Y))}$.