Electronic noise on a detector channel is modelled as a standard Gaussian voltage (in…

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 Monte Carlo integration and variance reduction

Problem

Electronic noise on a detector channel is modelled as a standard Gaussian voltage ZZ (in units of its own standard deviation). To estimate the rare-event probability P(Z>6)P(Z>6), importance sampling is used with a proposal density q(z)q(z), a unit-variance Gaussian centred at μ=6\mu = 6, so that the weight is w(z)=p(z)/q(z)=eμ2/2−μzw(z) = p(z)/q(z) = e^{\mu^2/2 - \mu z}, with pp the standard Gaussian density. The variance per sample of the weighted estimator can be written var=eμ2P(Z>2μ)−P(Z>μ)2.\mathrm{var} = e^{\mu^2}P(Z>2\mu) - P(Z>\mu)^2. Using P(Z>6)=9.87×10−10P(Z>6) = 9.87\times10^{-10} and P(Z>12)=1.78×10−33P(Z>12) = 1.78\times10^{-33}: ((a)) Find the variance per sample and the relative standard error of the importance-sampling estimate after N=1×104N = 1\times10^4 samples. ((b)) Find the number of plain Monte Carlo samples that would be needed for the same relative standard error, using RSE=(1−P)/(NP)\mathrm{RSE} = \sqrt{(1-P)/(NP)}.

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