A muon-decay experiment as in the chapter uses a shorter clock window

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

A muon-decay experiment as in the chapter uses a shorter clock window T=6.0 μsT = 6.0\,\mu\mathrm{s}. Idealise it as before: timing is exact, there is no background, and the N=400N = 400 recorded times are independent draws from the windowed density f(t;τ)=e−t/τ/[τ(1−e−T/τ)]f(t;\tau) = e^{-t/\tau}/[\tau(1-e^{-T/\tau})] on 0≤t≤T0\le t\le T. The variance of a recorded time is vT(τ)=τ2−T2e−T/τ/(1−e−T/τ)2v_T(\tau) = \tau^2 - T^2e^{-T/\tau}/(1-e^{-T/\tau})^2. The accepted value of the mean life is τ=2.197 μs\tau = 2.197\,\mu\mathrm{s}. ((a)) Find the bias of the sample mean tˉ\bar t as an estimate of τ\tau at this value of τ\tau. Express it in units of the standard error of tˉ\bar t for N=400N = 400, and say whether tˉ\bar t is consistent. ((b)) The experiment records tˉ=1.85 μs\bar t = 1.85\,\mu\mathrm{s}. Write the likelihood equation, show that it has exactly one root, and solve it by Newton's method starting from τ0=tˉ\tau_0 = \bar t. Give the first three iterates and the converged estimate τ^\hat\tau. ((c)) Find the Fisher information per recorded decay at τ^\hat\tau and the resulting error bar on τ^\hat\tau for N=400N = 400. State the result as τ^±σ\hat\tau \pm \sigma, and say how far the naive estimate tˉ\bar t lies from it in units of σ\sigma.

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