The number is to be estimated by Monte Carlo with uniform on and

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

The number

\ln2=\int_0^1\mathrm{d}u/(1+u)

is to be estimated by Monte Carlo with

U

uniform on

(0,1)

and

f(u)=1/(1+u)

. The control variate is

g(u)=u

, whose mean

\mu_g=\tfrac12

is known, so that each draw contributes

f(U)-c\,\big(g(U)-\mu_g\big)

to the average. ((a)) Show that

\mathrm{var}(f)=\tfrac12-(\ln2)^2

and

\mathrm{cov}(f,g)=1-\tfrac32\ln2

, evaluate both, and find the optimal coefficient

c^*

. ((b)) Find the correlation coefficient

\rho_{fg}

, and the factor by which the variance per draw at

c^*

is smaller than the plain variance

\mathrm{var}(f)

. ((c)) A student takes

c=1

, as in the example

f=e^{U}

g=1+U

of the text. Find the variance per draw at

c=1

and compare it with plain sampling. ((d)) Explain in two or three sentences why a negative

c^*

causes no difficulty.

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