A Try to swap between mean and partial derivatives on a product

AI Thread Summary
The discussion centers on the conditions under which mean and partial derivatives can be interchanged in product operations. It is established that these operations are swappable only if they are linear. A participant questions the validity of a specific equation presented, suggesting it may not hold true based on a brief review. The conversation emphasizes the need for clarity on the linearity of the operations involved. Overall, the exchange highlights the importance of verifying assumptions in mathematical derivations.
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I would like to be able to prove that we can swap the mean and partial derivatives on the defintion of a Fisher element matrix : this defintion involves the mean of a product of derivatives on Likelihood. I have also tried to formulate it with the ##chi^2## and the matrix of covariance of observables (noted "Cov" below). All of this is done in the goal that observable big "O" that I introduce is independent and so I have just to sum the extra elements calculated from "O".
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The operations are swappable if they're linear operations. Can you show that the operations are linear?
 
That said, I don't think this is a valid equation, as you seem to require, if my quick skimming of your post is correct:
$$\langle {\partial \chi^2 \over \partial \lambda_i \lambda_j} \rangle = \langle {\partial \chi^2 \over \partial \lambda_i} \rangle\langle {\partial \chi^2 \over \partial \lambda_j} \rangle$$
 
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