A detector's waiting time between successive decays from a steady source is…
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods
Mathematical Methods for Physics, Ch Probability, random variables and distributions
Problem
A detector's waiting time between successive decays from a steady source is exponentially distributed with mean τ=8ms, so each waiting time has mean τ and standard deviation τ. Consider the total elapsed time S until the fourth decay after a reference instant, S=T1+T2+T3+T4, where the Ti are the four independent waiting times between successive decays.
((a)) State the mean and standard deviation of S.
((b)) A Gaussian with the same mean and standard deviation as S is proposed as an approximation to its distribution. Using the fact that 95.4 per cent of a Gaussian's weight lies within two standard deviations of its mean, estimate the fraction of this Gaussian's weight lying at negative (physically impossible) times. Comment on what this shows about the reliability of the central limit theorem for a sum of only four terms.
((c)) As a question about the theorem's conditions rather than about decays (a Lorentzian takes negative values, so it cannot describe a waiting time), say what would change if S were instead the sum of four independent variables each drawn from the Lorentzian density f(x)=1/[πγ(1+(x/γ)2)] for some scale γ, and explain why in terms of the theorem's conditions.