Covariance

The covariance of two random variables $X,Y$ is defined as the central moment of $X,Y$ where $j=k=1$, i.e.,:

$$\text{cov}(X,Y)=E(X-E(X)\cdot (Y-E(Y)))=E(XY)-E(X)E(Y)$$

where $E(XY)$ is the correlation of $X,Y$. Then $\text{cov}(X,Y)=E(XY)$ if either of the random variables has a mean of 0.

Note that pairs of independent random variables have covariance zero. However, the inverse is not necessarily true, i.e., if two random variables have covariance zero, they may not necessarily be independent.