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Two fair six sided die are rolled, one called X the other Y. Evaluate the expected value of the difference of these two random variables, i.e., E((X-Y))

I'm not sure about this but I think the expected value of the difference should be 0. Since E(x) = 3.5 and E(y) = 3.5 so E(x-y) = 0

Am I interpreting this correctly?

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2 Answers 2

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Yes as stated, since $E[X-Y] = E[X]-E[Y]$, even if the dice were not independent but had the same expectations.

No if difference has to be non-negative, as in $E[|X-Y|]$.

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As stated, the expected difference of $X-Y$ will be 0. Not so if it is the expected absolute difference $|X-Y|$.

$$\begin{align} E(X-Y) &= E(E(X-Y\mid X)) \\ & = E(X-E(Y\mid X)) \\ & = E(X-E(Y)) \\ & = E(X)-E(Y) \\ & = 0 \\[3ex] E(|X-Y|) & = E(X-Y\mid X>Y)P(X>Y) + E(Y-X\mid X<Y)P(X<Y) + 0 P(X=Y) \\ & = \frac 5{12} \bigg(E(X-Y\mid X>Y) + E(Y-X\mid X<Y)\bigg) \\ & = \frac 5{6} E(X-Y\mid X>Y) \\ & =\frac 5{6} \frac 1 {15} \sum_{x=2}^{6} \sum_{y=1}^{x-1}(x-y) \\ & = \frac{35}{18} \\ & = 1.9\dot{\overline {4\,}} \end{align}$$

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