= Conditional pulse-energy mean given emission
{title2=$\mathbb E[U\mid E]$}
The \b[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 gives
$$
\mathbb E[U]=\Pr(E)\mathbb E[U\mid E].
$$
The 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.<Calibration Service for Laser Power and Energy at 248 nm by Rodney W. Leonhardt>[§6.2, Laser stability]{ref}
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