Lets say for we have a first dice and a second dice, with X representing the face of the first dice and Y representing the face of the second dice.
I know that E(X) = E(Y) = 3.5 $$ E(XY) = \sum_{(x,y)} xy P $$ Each side of the dice has a 1/6 chance to be rolled, every combination has a 1/36 chance.
What I can't seem to grasp is what $\sum_{(x,y)} xy P$ really means. What is the symbolic expansion of this, and what is ultimately E(XY)?
Since $X,Y$ are independent and uniformly distributed over $[1,\ldots,6]$, $$ \mathbb{P}[X=x,Y=y] = \mathbb{P}[X=x] \cdot \mathbb{P}[Y=y] = \frac1{36}. $$
Hence, $$ \mathbb{E}[XY] = \sum_{x = 1}^6 \sum_{y=1}^6 x y \mathbb{P}[X=x,Y=y] = \frac{1}{36} \sum_{x = 1}^6 \sum_{y=1}^6 x y $$ Can you finish this computation?