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Consider $X_1,X_2$ i.i.d. standard normal random variables(mean 0, variance 1). Are the random variables $Y=X_1+X_2$ and $Z=X_1-X_2$ dependent? I am not sure how to prove this one way or the other.

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    Use the fact that sums of jointly Gaussian random variables are Gaussian (exercise) and that two jointly Gaussian random variables are independent if and only if they have zero covariance (exercise). Note that the latter statement is false in general.2011-09-27
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    @QiaochuYuan A minor quibble: "two Gaussians are independent of and only if they have zero covariance" is not quite right since, as has been discussed elsewhere on math.stackexchange.com, joint Gaussianity is required for uncorrelated Gaussian random variables to be independent. Here of course, $Y$ and $Z$ _are_ jointly Gaussian and so the issue does not arise.2011-09-27
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    @Dilip: you're right, of course. Corrected.2011-09-27
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    We want $E(YZ)-E(Y)E(Z)$. But $YZ=X_1^2-X_2^2$.2011-09-27
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    So covariance is zero. So they are independent?2011-09-27
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    For *general* distributions, covariance $0$ does not imply independence. However, please see comments of Quiaochu Yuan and Dilip Sarwate.2011-09-27
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    @QiaochuYuan .... and if I might request yet another minor edit to your comment, "sums of Gaussians are Gaussian" is not quite correct; once again, joint Gaussianity (which fortunately holds in the question asked) is required.2011-09-27
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    @Dilip: my apologies once again. Edited.2011-09-27

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