Covariance

The covariance between two random variables $X$ and $Y$ is defined as $\mathrm{c}\mathrm{o}\mathrm{v}(X,Y)=\mathrm{E}((X-\mathrm{E}(X))(Y-\mathrm{E}(Y)))$.

Questions/things to explain

• A lot of explanations of covariance say things like "if the covariance is high, then the two variables vary together, so that when one is higher than average the other is as well". That sounds more like conditional expectation though, specifically $\mathrm{E}(Y-\mathrm{E}(Y)\mid X-\mathrm{E}(X))$.
• Explain difference (in units, range of values) with correlation.