For the covariance formula, how are subtraction and multiplication defined for real-valued random variables.

Question

Covariance is defined as $$E\left[(X-E[X])(Y-E[Y])\right]$$ If $X$ and $Y$ are real-valued random variables, how do we do multiplication and scalar subtraction on them?

Answer 1

If you have a function, $g:\Bbb R^2\to\Bbb R$, and continuously distributed real valued random variables $X,Y$, with joint probability distribution function $f_{X,Y}$, then:

$$\mathsf E(g(X,Y)) = \iint_{\Bbb R^2} g(x,y)\,f_{X,Y}(x,y)\operatorname d y\operatorname dx$$

Hence:

$$\begin{align}\mathsf E\Big(\big(X-\mathsf E(X)\big)\big(Y-\mathsf E(Y)\big)\Big) & = \iint_{\Bbb R^2} \big(x-\mathsf E(X)\big)\big(y-\mathsf E(Y)\big)\, f_{X,Y}(x,y)\operatorname d y\operatorname d x \\[2ex] = \mathsf E(XY)-\mathsf E(X)\mathsf E(Y) & = \iint_{\Bbb R^2} xy \, f_{X,Y}(x,y)\operatorname d y\operatorname d x-\int_\Bbb R x\,f_X(x)\operatorname d x\int_\Bbb R y\,f_Y(y)\operatorname d y\end{align}$$