A Geiger counter records the number of decays in each of equal

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Mathematical Methods for Physics, Ch Estimation, fitting and hypothesis testing

Problem

A Geiger counter records the number of decays njn_j in each of N=10N = 10 equal, non-overlapping intervals of length Δt=5 s\Delta t = 5\,\mathrm{s}, from a source whose decay rate λ\lambda is effectively constant over the measurement. Each count is an independent Poisson random variable with mean μ=λΔt\mu = \lambda\Delta t, so p(n;μ)=μne−μ/n!p(n;\mu) = \mu^n e^{-\mu}/n!. The ten counts are 12,8,15,9,11,14,10,13,9,1212, 8, 15, 9, 11, 14, 10, 13, 9, 12. ((a)) Write the likelihood of the ten counts as a function of λ\lambda and show that maximising ln⁡L\ln\mathcal{L} gives λ^=nˉ/Δt\hat\lambda = \bar n/\Delta t, where nˉ\bar n is the sample mean count. ((b)) Evaluate λ^\hat\lambda and quote its standard error, using the fact that the variance of a Poisson count equals its mean. ((c)) Least squares follows from a Gaussian likelihood. Explain briefly why treating these counts by least squares would only be an approximation, and judge whether a mean count near 1111 is large enough for that approximation to be fair.

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