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I was looking at the following question from "One Thousand Exercises in Probability" by Grimmett, page 25, question 16 (not homework just self-study):

Let $X$ and $Y$ be independent Bernoulli random variables with $p = 1/2$. Show that $X + Y$ and $|X − Y|$ are dependent though uncorrelated.

Now, the solution given on page 176 goes as follows:

$\mathrm{cov}(X+Y, |X-Y|) = E[(X+Y)\cdot(|X-Y|)] - E[X+Y]\cdot E[|X-Y|]$

I can work out that

$E[X+Y] = (0)(1/4) + (1)(1/2) + (2)(1/4) = 1$

and

$E[|X-Y|] = (0)(1/2) + (1)(1/2) = 1/2$

But I can't work out how to do

$E[(X+Y)\cdot(|X-Y|)]= ?$

The solution in the book for the covariance is:

$1/4 + 1/4 - 1(1/2) = 0$

To show dependence, the book provides the following solution:

$P(X+Y=0, |X-Y|=0) = 1/4$ is not the same as $P(X+Y=0)\cdot P(|X-Y|=0) = 1/8$

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    $\begingroup$ Just use the definition of expectation: $E[(X+Y)\cdot(|X-Y|)]=\sum_{y=0}^{1}\sum_{x=0}^{1}(x+y)\cdot|x-y|P(X=x)P(Y=y)$ $\endgroup$ – assumednormal Oct 27 '12 at 18:04
  • $\begingroup$ @Max That's it! I get the expected answer of $E[(X+Y)⋅(|X−Y|)] = 0 + 1/4 + 1/4 + 0 = 1/2$. Thanks! $\endgroup$ – Clair Crossupton Oct 27 '12 at 18:25
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Here you can do a separation of the cases because there are very few. Here we go:

$X = 0; Y = 0 \Rightarrow |X - Y| = 0;(X+Y) = 0;(X+Y)|X-Y| = 0$
$X = 1; Y = 0 \Rightarrow |X - Y| = 1;(X+Y) = 1;(X+Y)|X-Y| = 1$
$X = 0; Y = 1 \Rightarrow |X - Y| = 1;(X+Y) = 1;(X+Y)|X-Y| = 1$
$X = 1; Y = 1 \Rightarrow |X - Y| = 0;(X+Y) = 2;(X+Y)|X-Y| = 0$

By giving a $1/4$ weight to each of these cases, you should find that the expected value of $E\left((X+Y)|X-Y|\right)$ is $1/2$ and not $0$. But the covariance is

$$E\left((X+Y)|X-Y|\right) - E(X+Y)E(|X-Y|) = 1/2 - 1 \cdot 1/2 = 0.$$

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  • $\begingroup$ Excellent, thanks! I have edited my post for point 2, not sure what I should edit the title of the post to though? $\endgroup$ – Clair Crossupton Oct 27 '12 at 18:28

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