Given two fair dice, what is the expected value of their product?

Question

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?

Answer 1

Yes, your approach and answer are both correct. You could also take the average of the products of all possible cases. $${\bar x}=\frac {{\sum_{i=1}^6}{\sum_{j=1}^6}ij}{6\times 6}$$ $${\bar x}=\biggr(\frac {6\times7}{2}\biggr)^2\times \frac {1}{36}$$ $${\bar x}=12.25$$

Answer 2

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.