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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.

Missing-pulse bias in average-power inference arises when a nominal trigger rate is used as though every trigger emitted a pulse. With background removed and complete pulses counted, let $M$ be the number of triggers and $N$ the number of emissions during $T$. ThenThe quotient is mean energy per trigger, including failed triggers as zero-energy outcomes. It equals the conditional pulse-energy mean given emission only when every trigger emits, or after correction for the measured emission fraction. Historical calorimetric inference uses energy, injection time and pulse rate; those quantities must describe the same collection.^[ref]

For an original five-trigger record, suppose energies are $0,1,0,3,0\mu \mathrm{J}$. Mean energy per trigger is $0.8\mu \mathrm{J}$, while the two emitted pulses average $2\mu \mathrm{J}$. At a nominal $1\mathrm{kHz}$ trigger rate, their average power is $0.8\mathrm{mW}$. The actual emission rate is $400\mathrm{Hz}$, consistent with the same power and emitted-pulse mean.

A pulse monitor with a threshold above the weaker pulse could detect only the $3\mu \mathrm{J}$ event. Its detected-event average would then be $3\mu \mathrm{J}$, also different from the true emitted-pulse mean. Failure to detect is not automatically failure to emit. A calibrated pulse-resolving receiver and a trigger counter distinguish emitted, detected and commanded events. The average power of a laser pulse train article additionally treats continuous backgrounds and observation-window accounting.