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If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent?

What happens with the independence between $X+Y$ and $X-Y$ when $X \sim \mathrm{Normal}(\mu_x,\sigma_x^2)$ and $Y \sim \mathrm{Normal}(\mu_y,\sigma_y^2) $

Thank you

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    Given that $X$ and $Y$ are independent normal, $X+Y$ and $X-Y$ are independent if and only if $\sigma_X^2=\sigma_Y^2$.2012-10-03
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    You only need to prove their covariance is 0 since uncorrelation implies independence for normal distribution.2012-10-03
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    @Patrick: that is not true. If $X$ has a standard normal distribution and $Y=X$ when $|X| \gt k \approx 1.538$ and $Y=-X$ when $|X|\le k$ then correlation is $0$ and $Y$ also has a standard normal distribution but they are not independent.2012-10-03
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    @Henry Thank you for pointing it out. They need to be jointly normal distributed.2012-10-03

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