Covariance

Let $(\mathrm{\Omega },\mathrm{F},\mathbb{P})$ be a discrete probability space let $X:\mathrm{\Omega }\to S_x$ and $Y:\mathrm{\Omega }\to S_y$, be two random variables, where $S_x$ and $S_y$ are finite subsets of $\mathbb{R}$. Then the covariance of $X$ and $Y$ is defined as the mean of the following random variable:

In terms of $X$ and $Y$, the calculation for covariance is thus: