Is there a good reason to use covariance and not correlation?

Question

Correlation is a normalization of covariance by the standard deviation of each variable.

So is there a good reason (and example) when we should (and have) to use covariance and not correlation ?

Answer 1

Covariances are generally more easy to handle mathematically. If $X$ and $Y$ are random variables with expectation zero, $$ Cov(X,Y)=E[XY] $$ Whereas: $$ Corr(X,Y)=\frac{E[XY]}{\sqrt{E[X^2]}\sqrt{E[Y^2]}} $$ The covariance can be seen as an inner product (dot product) on the space of random variables with finite variance and expectation zero. Consequentially, it has many nice mathematical properties, and is important to study such random variables geometrically. An important example is that $X\mapsto Cov(X,Y)$ is linear, but $X\mapsto Corr(X,Y)$ is not linear.