Question: Given two fair dice, what is the expected value of their product?
My attempt:
Let $X_1$ and $X_2$ be scores by first and second die respectively.
Note that $X_1$ and $X_2$ are independent.
Then
$$E(X_1X_2) = E(X_1)E(X_2) = 3.5^2 = 12.25.$$
Is my calculation correct?
Best Answer
The expectation of a product of independent random variables is equal to the product of their expectations.
$$E[X_1 X_2] = \sum_{x_1} \sum_{x_2} x_1 x_2 p(X_1 = x_1 {\rm \ and\ } X_2=x_2)$$ By independence of $X_1$ and $X_2$, $$p(X_1 = x_1 {\rm \ and\ } X_2=x_2) = p(X1=x1) p(X_2=x2)$$ so $$E[X_1 X_2] = \sum_{x_1} \sum_{x_2} x_1 x_2 p(X_1 = x_1) p(X_2=x_2) \\ = \sum_{x_1} x_1 p_1(X_1 = x_1) \cdot \sum_{x_2} x_2 p(X2=x2) \\ \sum_{x_1} x_1 p(x_1) E[X_2] \\ E[X_1]E[X_2]$$
So you are correct with your calculations as long as the rolls are independent.