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Error propagation in ELUT correction #248

@samaloney

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@samaloney

So essentially we are doing

$$ c' = \sum_{i=0}^{N} w_i c_i $$

where $w_i$ is the over lap between the science and actual energy bin but this is simplified as

$$ c' = \sum_{i=n+1}^{N-1} 1 c_i + \sum_{j=\{0,N\}} w_j c_j $$

because the internal bins are summed so there edge position doesn't matter so the correction can be ignored. So then the error should be

$$ {\sigma`}^2 = \sum_{i=n+1}^{N-1} 1^2 c_i + \sum_{j= \{ 0,N \} } w_j^2 c_j $$

Should the energy_corr_factor_low and energy_corr_factor_high be squared (not sure if counts_error is the $\sigma$ or $\sigma^2$ so may need to be squared too?)

counts[energy_ind[0],*,*] *= energy_corr_factor_low
counts[energy_ind[-1],*,*] *= energy_corr_factor_high
counts_error[energy_ind[0],*,*] *= energy_corr_factor_low
counts_error[energy_ind[-1],*,*] *= energy_corr_factor_high

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