Lecture 20: Correlation and Covariance

If X and Y are random variables defined on the same probability space $(\Omega, {\cal F}, \Pr)$ the covarariance of X and Y, denoted by ${\rm Cov}(X,Y)$ is defined by

$${\rm Cov}(X,Y) = E[(X-E[X])(Y-E[Y]$$

and the covariance coefficient is of X and Y, denoted by $\rho(X,Y)$ is defined by

$$\rho(X,Y)\sqrt{Var(X)}\sqrt{Var(Y)} = {\rm Cov}(X,Y).$$

It follows from the Cauchy-Schartz inequality that $-1\leq \rho(X,Y) \leq 1$.Of course, covariance is undefined if the variance of X or Y is . However, in this case either X or Y is constant, and X and Y are independent of each other.