Someone argues: " unstable nuclei each decay independently

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 Random processes in continuous time

Problem

Someone argues: "N0=40N_0 = 40 unstable nuclei each decay independently, with survival probability p=e−t/τp = e^{-t/\tau} by time tt. The number N(t)N(t) still present is well approximated by a Poisson distribution with mean ⟨N(t)⟩=N0p\langle N(t)\rangle = N_0 p. At t=τt = \tau, p=e−1=0.368p = e^{-1} = 0.368, so ⟨N(τ)⟩=40×0.368=14.7\langle N(\tau)\rangle = 40\times0.368 = 14.7, and because a Poisson distribution has variance equal to its mean, var(N(τ))=14.7{\rm var}(N(\tau)) = 14.7 as well." ((a)) Identify the error in this reasoning. ((b)) Give the correct value of var(N(τ)){\rm var}(N(\tau)).

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