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This is an experimental physics problem:

Say I have 3 random variables $P_1, P_2, P_3 $ such as :

$\Delta P_i=\Delta_Q P_i +\Delta_0P_i \ \ \forall i$, where $\Delta$ is the variance, this equations means that the variance of $P_i$ is the sum of two variances (experimental noises are independent and added quadratically).

These noises are different in nature, they are due to two different physical effects, the first one indexed $Q$ correlates the three variables in the sense that :

${\rm cov_Q}(P_i,P_j)=\overline{P_i}\times\overline{P_j}$, where the bar denotes the average. Here ${\rm cov_Q}(P_i,P_j)$ is the covariance of the two variables would have, if they were only affected by the physical effect Q (I hope it makes sense).

The other one does not correlate the three variables : ${\rm cov_0}(P_i,P_j)=0$

Additional info : $\overline{P_1+P_2+P_3}=0$

Question: Is it possible to extract the value of $\Delta_Q P_i$ or equivalently $\Delta_0 P_i$ knowing that I have experimental access to all $\overline{P_i}$ and $\Delta P_i$ ?

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    $\begingroup$ Although I struggle with your definition of the variance $\Delta P_i$ -- it would be helpful to write the result as a function, $f(P_1, P_2, P_3)$ -- the answer to your question is: Check out variance component analysis. It's a rather complicated theory, so I suggest you start with the simples model: fixed effect balanced design. $\endgroup$ – Semoi Feb 7 at 15:18
  • $\begingroup$ thank you, I will check it out ! $\endgroup$ – yfs Feb 7 at 15:38

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