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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. – Qiaochu Yuan Sep 27 '11 at 18:33
@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, 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. – Dilip Sarwate Sep 27 '11 at 18:43
@Dilip: you're right, of course. Corrected. – Qiaochu Yuan Sep 27 '11 at 18:45
We want $E(YZ)-E(Y)E(Z)$. But $YZ=X_1^2-X_2^2$. – André Nicolas Sep 27 '11 at 18:57
So covariance is zero. So they are independent? – DumbQuestion Sep 27 '11 at 19:04

$X_1$ and $X_2$ are independent standard normals, so $(X_1, X_2)$ has rotationally symmetric density, namely $$ {1 \over 2\pi} \exp(-(x_1^2 + x_2^2)/2). $$ If you change coordinates with $u = (x_1 + x_2)/\sqrt{2}, v = (x_1 - x_2)/\sqrt{2}$ (so the change from $(x_1, x_2)$ to $(u,v)$ is area-preserving) then this becomes $$ {1 \over 2\pi} \exp(-(u^2+v^2)/2). $$ That is, the random variables $U = (X_1 + X_2)/\sqrt{2}$ and $V = (X_1 - X_2)/\sqrt{2}$ are also independent standard normals. Your random variables are $Y = U \sqrt{2}$ and $Z = V \sqrt{2}$, so they're independent normals with mean 0 and SD $\sqrt{2}$.

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Corrected some typos. Please check the result suits you. – Did Sep 28 '11 at 8:56
Thanks. This is what happens when I write quickly. – Michael Lugo Sep 28 '11 at 15:26

If you are familiar with the concept of characteristic function, it is easiest to compute characteristic function for $(Y, Z)$. For independent variables, the characteristic function would factor into a product:

$$ \begin{eqnarray} \mathbb{E}\left( \exp( i t_1 Y + i t_2 Z ) \right) &=& \mathbb{E}\left( \exp( i (t_1+t_2) X_1 + i (t_1-t_2) X_2 ) \right) \\ & = & \exp\left( -\frac{1}{2} \left(t_1+t_2\right)^2 \right) \cdot \exp\left( -\frac{1}{2} \left(t_1-t_2\right)^2 \right) \\ &=& \exp\left( -t_1^2 \right) \cdot \exp \left(-t_2^2 \right) \end{eqnarray} $$

Hence the $Y$ and $Z$ are independent normal with mean 0 and standard deviation of $\sqrt{2}$.

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