A comparator on a photodiode channel fires when the noise voltage

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

Z

, a standard Gaussian in units of its rms value, exceeds a threshold of

4

. The false-alarm probability

P=P(Z>4)=3.167124\times10^{-5}

is to be estimated by importance sampling, with

f=1

for

x>4

and

0

otherwise,

p(x)=e^{-x^2/2}/\sqrt{2\pi}

and the shifted exponential proposal

q_\lambda(x)=\lambda\,e^{-\lambda(x-4)},\qquad x>4,

where the rate

\lambda>0

is to be chosen. Every draw from

q_\lambda

has

f=1

, so the estimate is

N^{-1}\sum_iw(X_i)

with

w=p/q_\lambda

. Evaluate Gaussian tails

P(Z>t)

with software. ((a)) Show that

q_\lambda/p\ge c

for all

x>4

with

c=\lambda\sqrt{2\pi}\,e^{4\lambda-\lambda^2/2}

, and conclude that the variance of the estimator is finite for every

\lambda>0

. ((b)) Show that the second moment

M(\lambda)=\int_4^\infty p^2/q_\lambda\,\mathrm{d}x

equals

M(\lambda)=\frac{e^{\lambda^2/4-4\lambda}}{2\sqrt\pi\,\lambda}\,P\!\left(Z>4\sqrt2-\frac{\lambda}{\sqrt2}\right),

and find the

\lambda^*

that minimises it, to three significant figures. ((c)) Find the relative error of the estimate from

N=10^4

draws, in per cent, at

\lambda^*

, at

\lambda=4

, and for the shifted unit Gaussian

q(x)=e^{-(x-4)^2/2}/\sqrt{2\pi}

, for which

M=e^{16}P(Z>8)

with

P(Z>8)=6.22096\times10^{-16}

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