A photon-counting detector records the number of photons arriving in each of independent

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 photon-counting detector records the number of photons arriving in each of N=200N=200 independent, equal-length time intervals. In each interval the count nn is Poisson distributed with the same unknown mean λ\lambda, a pure number of counts, and the 200200 recorded counts average to nˉ=4.0\bar n = 4.0. ((a)) Show that the Fisher information carried by a single interval is I(λ)=1/λI(\lambda) = 1/\lambda. ((b)) The result is to be quoted as θ=λ\theta = \sqrt\lambda. The score for θ\theta is the score for λ\lambda multiplied by dλ/dθ\mathrm{d}\lambda/\mathrm{d}\theta. Find the information per interval I(θ)I(\theta), show that it does not depend on λ\lambda, and state the Cramér-Rao bound on the variance of an unbiased estimator of θ\theta built from all N=200N=200 intervals. ((c)) The estimate is θ^=nˉ\hat\theta = \sqrt{\bar n}, and var(nˉ)=λ/N\mathrm{var}(\bar n) = \lambda/N. Propagate this variance to first order to find var(θ^)\mathrm{var}(\hat\theta), and compare it with the bound from part (b).

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