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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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    @Henry Thank you for pointing it out. They need to be jointly normal distributed.2012-10-03

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HINT:

  1. Independent Gussian random variables makes Gaussian random vector.
  2. Affine transform $Y=A X + b$ of Gaussian random vector $X$ is Gaussian.
  3. Distribution of Gaussian random vector is determined by its mean vector, and covariance matrix.
  4. If components $X_i$ and $X_j$ of the Gaussian random vector are independent, then $\mathbb{Cov}(X_i, X_j) = 0$.

Combining facts given above, it follows that evaluation of $\mathbb{Cov}(X+Y,X-Y)$ will help establish the result needed.

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    @did Thanks. $A$dded.2012-10-03