Getting paid the product of two dice

Question

You roll two fair six-sided dice and are paid the product of the two faces in dollars (so rolling a 4 and a 3 pays \12). What is the fair value of this game?

Accepted Answer

The key fact: the two dice are independent, and for independent variables the expectation of a product is the product of the expectations: E[XY] = E[X]\,E[Y]. Each die has mean E[X] = \frac{1+2+3+4+5+6}{6} = 3.5. So E[XY] = 3.5 	imes 3.5 = 12.25. The fair value is \12.25. Takeaway. E[XY] = E[X]E[Y] holds because the dice are independent, it is not a general rule. For correlated variables the product's expectation also picks up a covariance term, E[XY] = E[X]E[Y] + 	ext{Cov}(X,Y), which vanishes only under independence (or zero correlation).

Getting paid the product of two dice

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