= Separating detector noise from pulse-energy variance
{title2=$\operatorname{Var}(X)=\operatorname{Var}(U)+\operatorname{Var}(n)+2\operatorname{Cov}(U,n)$}
\b[Separating detector noise from pulse-energy variance] estimates source variability using a model of the readout's own fluctuations. In energy-calibrated units, write $X=U+n$. The exact identity is
$$
\operatorname{Var}(X)=\operatorname{Var}(U)+\operatorname{Var}(n)+2\operatorname{Cov}(U,n).
$$
Only when the covariance vanishes can a separately estimated noise variance be subtracted directly. Covariance terms are part of measurement-model uncertainty propagation, rather than an optional correction.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[AppendixA, equationA-3]{ref}
For an original independent-noise example, suppose the observed energy standard deviation is $0.50\,\mathrm{nJ}$ and a matched noise-only measurement gives $0.30\,\mathrm{nJ}$. The inferred source standard deviation is $\sqrt{0.50^2-0.30^2}=0.40\,\mathrm{nJ}$, not $0.50-0.30=0.20\,\mathrm{nJ}$. The source contributes $64\%$ of the observed variance in this stipulated model.
A noise-only record should use comparable gain, integration gate and reset conditions. <Background subtraction> removes an estimated mean, not the random variance remaining around it. <Amplifier clipping>, unmatched <integrator reset> or signal-dependent detector noise can invalidate an additive independent-noise model. A negative variance difference is not a physical negative source variance; it can indicate estimation uncertainty or mismatched assumptions.
Splitting pulses between two independently characterized readouts can provide a <two-channel covariance estimate of pulse-energy variance> under additional noise assumptions. Shared electronics noise and common drift must still be assessed; independence cannot be established just by using two detectors.
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