Expected value when rolling dice

Question

If I roll two fair dice and let $Y$ be the number obtained by multipling the scores, the expected value of $Y$ is $E[Y]=\dfrac{49}{4}$. Let $X$ be the score obtained by rolling a single fair die. Then $E[X]=\dfrac{7}{2}$. Is it a coincidence that $E[Y]=(E[X])^2$?

Answer 1

Theorem $1$. If $A$ and $B$ are independent discrete random variables, then $$E[A]\times E[B]=E[AB].$$

Proof.

$$\begin{aligned}E[AB]&=\sum_{a,b}abP(A=a \cap B=b)\\ &=\sum_{a,b}abP(A=a)P(B=b) \quad\quad\quad \text{ (since $A$ and $B$ are independent)}\\ &=\left(\sum_a aP(A=a)\right)\left(\sum_b bP(B=b)\right)\\ &=E[A]E[B] \end{aligned}$$

Theorem $2$. If $A$ and $B$ are independent continuous random variables, then $$E[A]\times E[B]=E[AB].$$

Proof.

See here.