Is XY the joint distribution over X and Y?

Question

The definition of covariance is

$Cov(X,Y) = E[XY] - E[X]E[Y]$

I can't wrap my head around what $XY$ is supposed to be. I suspected it to be the joint distribution over $X$ and $Y$ but I could not find this notation in the respective wikipedia article . If it is not the joint distribution, what is it then?

Answer 1

$E[XY]$ is the expectation of the product of the two random variables $X$ and $Y$. If they are independent then $E[XY]=E[X]E[Y]$ so

$$\operatorname {Cov}(X,Y)=0.$$

Answer 2

If $X:\Omega \to \mathcal{R}$ and $Y:\Omega \to \mathcal{R}$ are two random variables, then $XY: \Omega \to \mathcal{R}$ is another random variable given by $$XY(\omega)=X(\omega)Y(\omega)$$ the product of $X$ and $Y$. In order to find distribution of $XY$ you need to know the joint distribution of random vector $(X,Y)$.