10.6 Exercises on Chapter 10
1. Show that in importance sampling the choice
minimizes 
even in cases where f(x) is not of constant sign.
even in cases where f(x) is not of constant sign.
2. Suppose that has a Cauchy distribution. It is easily shown that , but we will consider Monte Carlo methods of evaluating this probability.
a. Show that if k is the number of values taken from a random sample of size n with a Cauchy distribution, then k/n is an estimate with variance 0.125 802 7/n.
b. Let p(x)=2/x2, so that 
. Show that if 
is uniformly distributed over the unit interval then y=2/x has the density p(x) and that all values of y satisfy 
and hence that
. Show that if is uniformly distributed over the unit interval then y=2/x has the density p(x) and that all values of y satisfy and hence that
gives an estimate of 
by importance sampling.
by importance sampling.
c. Deduce that if 
are independent U(0, ...
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