Photon counts from a steady source are recorded in intervals of fixed length

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

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

Problem

Photon counts nn from a steady source are recorded in intervals of fixed length Δt=1 s\Delta t = 1\ \mathrm{s}. The source emits at a rate λ\lambda (in s−1\mathrm{s}^{-1}), so the count in one interval is Poisson distributed with mean λΔt\lambda\Delta t, P(n;λ)=e−λΔt(λΔt)n/n!P(n;\lambda) = e^{-\lambda\Delta t}(\lambda\Delta t)^n/n!. An experiment records N=500N = 500 independent intervals. For the purpose of this design calculation, take the true rate as λ=8 s−1\lambda = 8\,\mathrm{s}^{-1}. ((a)) Starting from ln⁡L(λ;n)=−λΔt+nln⁡(λΔt)−ln⁡n!\ln L(\lambda;n) = -\lambda\Delta t + n\ln(\lambda\Delta t) - \ln n! for one interval, derive the Fisher information per interval, I(λ)I(\lambda), and evaluate it with its units at λ=8 s−1\lambda = 8\,\mathrm{s}^{-1}. ((b)) Compute the Cramér-Rao bound on the variance of any unbiased estimator of λ\lambda from the NN intervals. ((c)) The estimate λ^=nˉ/Δt\hat\lambda = \bar n/\Delta t, the mean count per interval divided by Δt\Delta t, is unbiased with variance λ/(NΔt)\lambda/(N\Delta t), since each count has variance λΔt\lambda\Delta t. Compare this with the bound and state whether the sample mean is efficient.

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