What does the values of covariance matrix tell us?

Question

Assume we have the following covariance matrix:

cov(x,y) = [var(x),    cov(x,y)  = [4 3
            cov(y,x),  cov(y) ]     3 4]

• What can we tell about the value of 3 ? (cov(x,y) = 3 ?)
• Can we infer that "one change of x, the y will change 3 times" ?
• Is there a meaning for the value 3 ? or we just used it's sign (positive or negative) ?

Answer 1

The covariance is a measure of linear dependence between $X$ and $Y$. In particular, you can infer that

$$\rho_{XY}=\frac{\mathbb{Cov}(X,Y)}{\sigma_X\cdot \sigma_Y}=75\%$$

...there is a positive linear dependence of 75% between the two rv's