An integral is estimated by plain Monte Carlo sampling: points are drawn uniformly on…

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

An integral ∫01f(x) dx\int_0^1 f(x)\,\mathrm dx is estimated by plain Monte Carlo sampling: N=10 000N = 10\,000 points xix_i are drawn uniformly on [0,1][0,1] and the values f(xi)f(x_i) are averaged. The sample variance of the 10 00010\,000 values f(xi)f(x_i) is s2=0.19s^2 = 0.19. ((a)) State the standard error of the Monte Carlo estimate of the integral. ((b)) How many samples in total would be needed to halve this standard error?

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