Covariance

Covariance is built from expectation on a probability space and generalizes variance, since $\operatorname{Cov}(X,X)=\operatorname{Var}(X)$. If $X$ and $Y$ are independent and have finite second moments, then $\operatorname{Cov}(X,Y)=0$ (but the converse need not hold).

• If $X=\mathbf{1}_A$ and $Y=\mathbf{1}_B$ are indicator random variables of events $A,B$, then $\operatorname{Cov}(X,Y)=\mathbb{P}(A\cap B)-\mathbb{P}(A)\mathbb{P}(B)$ in terms of event probabilities.
• If $Y=aX+b$ for constants $a,b$ and $X$ has finite second moment, then $\operatorname{Cov}(X,Y)=a\,\operatorname{Var}(X)$.
• If $X$ and $Y$ are independent standard normal random variables, then $\operatorname{Cov}(X,Y)=0$.