Covariance

A covariance is the quantity $\operatorname{Cov}(X,Y)=\mathbb{E}\!\left[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])\right]=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]$ associated to two random variables $X$ and $Y$ with finite second moments.

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).

Examples:

• 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 $\operatorname{Cov}(X,Y)=0$.