Conditional photon-number bias (${\bar{n}}_{-}=\mu -1+\mathrm{Var}N/\mu$)

A subtraction event weights initial occupation $n$ by $n$ before removing one photon. Thus its conditional output mean is $⟨N(N-1)⟩/\mu$, giving the displayed identity for $\mu >0$. If the initial variance exceeds its mean, this biased selection can offset the removed photon. The unconditional ensemble includes unsuccessful events and has a different energy balance.