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Let X be the number appearing on the first die when two fair dice are rolled, and let Y be the sum of the numbers appearing on the two dice. Show that E(X)E(Y) does not equal to E(XY).

I found E(X) and E(Y) but I don't know how to find E(XY). The answer in the textbook says that E(XY)= 329/12. How is this the case?

aralc
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1 Answers1

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Here is a reasonably efficient way to find $E(XY)$. Let $W$ be the number on the second die. Then $Y=X+W$ so $XY=X^2+XW$. It follows that $E(XY)=E(X^2)+E(XW)$.

By independence, $E(XW)=E(X)E(W)$. Finally, to find $E(X^2)$, we can compute directly. We have $E(X^2)=\frac{1}{6}(1^2+2^2+3^2+4^2+5^2+6^2)$.

Now put the pieces together. If we wish, we can compute $1^2+2^2+\cdots+6^2$ by the formula for the sum of the first $n$ squares, but direct calculation is really no harder.

Remark: Alternately, we can find $E(XY)$ by finding $\Pr(XY)=k$ for all possible values of $k$, and using the ordinary formula for expectation. In principle this is straightforward, in practice excessively tedious.

André Nicolas
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  • I don't really understand what XY means. Is it what X and Y have in common? How would I find this? – aralc Aug 01 '14 at 16:44
  • @aralc : $XY$ is what you get when you multiply $X$ and $Y$. ${}\qquad{}$ – Michael Hardy Aug 01 '14 at 16:52
  • @aralc: Algebraic manipulation of random variables, particularly of sums and products, will be essential in further work. The random variable $XY$ is mentioned explicitly in the question. To answer the question, either by brute force or as above, one needs to know what it means. – André Nicolas Aug 01 '14 at 16:59