= Finite-exposure correction to fluctuation force calibration
{title2=$\langle\bar x^2\rangle/\langle x^2\rangle=2(s-1+e^{-s})/s^2$}
For an equilibrium overdamped harmonic coordinate with covariance $\langle x(t)x(0)\rangle=V e^{-|t|/\tau}$, a camera exposure $\Delta$ records the average over that interval. Integrating the covariance twice gives $\operatorname{Var}(\bar x)=2V\Delta^{-2}\int_0^\Delta(\Delta-u)e^{-u/\tau}du$. The displayed reduction follows with $s=\Delta/\tau$. In the tether calibration $F=k_BTh/V$, 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.
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