Finite-exposure correction to fluctuation force calibration ($⟨{\bar{x}}^2⟩/⟨x^2⟩=2(s-1+e^{-s})/s^2$)

For an equilibrium overdamped harmonic coordinate with covariance $⟨x(t)x(0)⟩=V{e}^{-|t|/\tau }$, a camera exposure $\Delta$ records the average over that interval. Integrating the covariance twice gives $\mathrm{Var}(\bar{x})=2V{\Delta }^{-2}{\int }_0^{\Delta }(\Delta -u)e^{-u/\tau }du$. The displayed reduction follows with $s=\Delta /\tau$. In the tether calibration $F=k_{B}Th/V$, ignoring exposure averaging overestimates the inferred force. Independent localization-noise variance instead adds to the measured variance and biases it in the opposite direction. Other filters require their actual response functions.