The integral is estimated from antithetic pairs with uniform 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

The integral

I=\int_0^1f(u)\,\mathrm{d}u

is estimated from antithetic pairs

Y=\tfrac12\big[f(U)+f(1-U)\big]

with

U

uniform on

(0,1)

, and costs are compared at equal numbers of evaluations of

f

. Three integrands are considered:

f_1(u)=u^2

f_2(u)=\sin\pi u

and

f_3(u)=(u-\tfrac12)^3

. Let

\rho_{\rm a}

be the correlation coefficient of

f(U)

and

f(1-U)

. ((a)) Before calculating anything, predict for each function whether antithetic pairs lower the variance at equal cost, raise it or leave it unchanged, with a one-line reason. ((b)) For

f_1

, find

\rho_{\rm a}

and the factor by which the variance at equal cost is reduced. ((c)) For

f_2

, find

\rho_{\rm a}

and the same factor. ((d)) For

f_3

, find

\rho_{\rm a}

and the variance of

Y

, and say why the factor

1/(1+\rho_{\rm a})

cannot be quoted as a finite number here.

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