Photons scattered once inside a thin foil emerge with a scattering angle that

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

Error, Chance and the Machine, Ch Monte Carlo integration and variance reduction

Problem

Photons scattered once inside a thin foil emerge with a scattering angle θ\theta that, for this estimate, is sampled uniformly on [0,π][0,\pi]. The quantity of interest is ⟨sin⁡2θ⟩=1π∫0πsin⁡2θ dθ\langle \sin^2\theta\rangle = \frac{1}{\pi}\int_0^\pi \sin^2\theta\,\mathrm{d}\theta, estimated from NN samples θi\theta_i by I^=1N∑i=1Nsin⁡2θi\hat I = \frac1N\sum_{i=1}^N \sin^2\theta_i. ((a)) Find var(sin⁡2θ)\mathrm{var}(\sin^2\theta) for θ\theta uniform on [0,π][0,\pi]. ((b)) Find the smallest NN such that the standard error of I^\hat I is no more than 1 per cent of the true mean value ⟨sin⁡2θ⟩=12\langle\sin^2\theta\rangle = \tfrac12.

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