Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 338 / 3 / b / ii

Put $Q=R_{Q}T$ and $B=R_{B}T$ in the root-mean-square accuracy measure:For a fixed uncorrected $f\ne 1$ and $R_B>0$, the squared bias eventually dominates the shot-noise terms. ThereforeThe PDF omits the absolute value. Its expression is the positive accuracy ratio only if $f<1$; for $f>1$ the residual background changes sign, but an error magnitude and a signal-to-noise ratio in photon counting remain nonnegative. The printed signed formula can instead be read as source divided by signed bias.

For exact background matching $f=1$, there is no systematic-error signal-to-noise ceiling: $Z=R_Q\sqrt{T/(R_Q+2R_B)}$ grows without bound in this idealized model. With $R_B=0$ the same conclusion holds. Limits in which $f$ itself approaches one with exposure time are different from the fixed-mismatch limit used here.