Covariance

Covariance measures the direction of the linear relationship between two random variables.

$$\text{Cov}(X,Y)\equiv {\sigma }_{XY}\equiv E\left[(X-E[X])(Y-E[Y])\right]E[XY]-E[X]E[Y]$$

• $\text{Cov}(X,X)=\text{Var}(X)$
• $\text{Cov}(X,Y)>0⟹$ Variables move in the same direction.
• $\text{Cov}(X,Y)<0⟹$ Variables move in opposite directions.
• $X$ and $Y$ are independent $⟹\text{Cov}(X,Y)=0$