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I have two distinct data samples($A$ and $B$), and to each one a gaussian is fitted. I then evaluate the product $S = \sigma_A * \sigma_B$ ($\sigma_A$ and $\sigma_B$ and their errors are obtained from fit procedure).

If I assume $\sigma_A$ and $\sigma_B$ are uncorrelated, I can easily propagated the error on the product ($S$) based on the estimates of $\sigma_A$ and $\sigma_B$ errors.

But what would be the correct way to propagate the error to include a possible correlation between $\sigma_A$ and $\sigma_B$?

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Let me say that I don't know, but I can make an educated first guess. (Hopefully this might help / provide some food for thought until someone else can answer w/ the derivation you need; then I'll delete this comment). You would put the correlation matrix b/t the variables. That is, $S=\sigma_A*\Sigma_{\sigma_A,\sigma_B}*\sigma_B$. NB, that when the errors are uncorrelated, this will reduce the the equation you listed, as the correlation matrix will be the identity matrix. – gung Aug 3 '12 at 13:41

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