 Conditional pulse-energy mean given emission

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]