A fluorescent dye has decay lifetime

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Estimators and the information in data

Problem

A fluorescent dye has decay lifetime τ=15.0 ns\tau = 15.0\,\mathrm{ns}. A detector records the arrival time of each emitted photon after excitation, but only within a window of length TT starting at t=0t=0; photons arriving after TT are discarded. With x=T/τx = T/\tau, the fraction of the untruncated Fisher information about τ\tau that survives the truncation is I(T)I(∞)=1−x2e−x(1−e−x)2,I(∞)=1τ2.\frac{I(T)}{I(\infty)} = 1 - \frac{x^2 e^{-x}}{(1-e^{-x})^2}, \qquad I(\infty) = \frac{1}{\tau^2}. ((a)) Evaluate this fraction for windows T=15T = 15, 3030 and 60 ns60\,\mathrm{ns}. ((b)) For an experiment recording N=500N = 500 photons within each such window, use the fraction from part (a) to give the Cramér-Rao bound on the standard deviation of τ^\hat\tau for each window length. ((c)) State in one sentence how the precision changes as the window is lengthened, and why this model (which ignores background and dead time) always favours a longer window.

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