Definition
\[\mathrm{cov}(X,Y)=\mathbb{E}\left[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])\right]\]
Properties
\[\mathrm{cov}(X,X)=\mathrm{var}(X)\]
\[\mathrm{cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\]
The Correlation Coefficient
\[\rho(X,Y)=\dfrac{\mathrm{cov}(X,Y)}{\sigma_X\sigma_Y}\]
\[-1\leq\rho\leq 1\]