= Bias of pooled flat-field ratio gain
{title2=$g_{\rm naive}/g=1/(\langle N\rangle\langle1/N\rangle)$}
For stable positive <detector pixel> means $N_i$ and uniform <detector conversion gain> $g$, high-count independent <Poisson distribution> noise gives $\operatorname{Var}(A_i/B_i)\simeq2/(gN_i)$. Thus substituting a global mean in the equal-signal formula yields $g_{\rm naive}=g/[\langle N_i\rangle\langle1/N_i\rangle]\le g$, by the <Jensen inequality>. Equality holds for uniform means. The <flat-field correction> pattern cancels from the ratio mean but not from the signal-dependent noise. Local signal bins, or a <variance> fit after normalization of differences by the square root of their local mean, avoid this leading bias.
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