Conditional pulse-energy mean given emission ($\mathbb{E}[U\mid E]$)

The conditional pulse-energy mean given emission averages over emitted optical pulses rather than all commands or all detected events. If $E$ denotes an emission event and energy is zero when no pulse is emitted, the conditional expectation identity givesThe left side is energy per trigger. The conditional mean is energy per emitted pulse. Missing-pulse bias in average-power inference explains the corresponding rate denominator.

For an original distribution, let emission probability be $0.75$. Conditional on emission, energies $2$ and $6\mathrm{nJ}$ are equally likely. The emitted-pulse mean is $4\mathrm{nJ}$, whereas the per-trigger mean is $3\mathrm{nJ}$. At $2\mathrm{MHz}$ nominal triggering, average pulse power is $6\mathrm{mW}$; the actual $1.5\mathrm{MHz}$ emission rate gives the same power when multiplied by $4\mathrm{nJ}$.

Now let a perfect threshold receiver detect only energies above $4\mathrm{nJ}$. Its detected-event mean is $6\mathrm{nJ}$ and its event probability per trigger is $0.375$. This is detection-threshold selection bias in pulse-energy means: the detector conditions on a narrower event than emission. The undetected records include both failed emissions and the weaker emitted pulses.

A calibrated pulse monitor, trigger counter and known detection response help distinguish the denominators. A total-energy measurement alone need not identify the missing weak-pulse distribution. Historical calorimetric inference combines total energy, injection time and pulse rate, so the rate must match the population whose mean is reported.^[ref]

Detection-threshold selection bias in pulse-energy means occurs when detected pulses have a different energy distribution from all emitted pulses. A detector conditions the reported mean on detection, not merely on emission. Let $U$ be emitted-pulse energy with finite mean and let $s(u)=\mathrm{Pr}(D\mid U=u,E)$ be detection probability conditional on emission $E$. ThenThe denominator must be positive. An ideal energy threshold has $s(u)=1$ above threshold and zero below. Its nondecreasing selection preferentially retains stronger pulses; false triggers or selection on another variable require a different model.

For an original distribution, emitted energies are $1$, $3$ and $7\mathrm{nJ}$ with probabilities $0.25$, $0.50$ and $0.25$. Their conditional pulse-energy mean given emission is $3.5\mathrm{nJ}$. A perfect receiver accepting only energies above $2\mathrm{nJ}$ detects $75\%$ of emitted pulses and reportsThe detected-event mean is about $23.8\%$ too high for the emitted-pulse mean, even with perfect energy measurements on every accepted event.

A trigger counter, a lower-threshold reference monitor and a threshold sweep can distinguish missing detections from missing emissions. Historical calorimetric inference uses total energy, exposure time and pulse rate; the rate must describe the population whose mean is requested.^[ref] This selection example is an independent probability model, not a reconstruction of that experiment.

Known positive detection probabilities can support inverse probability weighting. A hard threshold with zero probability below it leaves that part of the energy distribution unidentified from detected events alone. Reporting the acceptance fraction as well as the detected-event mean makes the conditioning explicit; it does not recover the unseen weak-pulse energies.