556 11. SAMPLING METHODS
Exercises
11.1 ( ) www Show that the finite sample estimatorf̂defined by (11.2) has mean
equal toE[f]and variance given by (11.3).
11.2 ( ) Suppose thatzis a random variable with uniform distribution over(0,1)and
that we transformzusingy=h−^1 (z)whereh(y)is given by (11.6). Show thaty
has the distributionp(y).
11.3 ( ) Given a random variablezthat is uniformly distributed over(0,1), find a trans-
formationy=f(z)such thatyhas a Cauchy distribution given by (11.8).
11.4 ( ) Suppose thatz 1 andz 2 are uniformly distributed over the unit circle, as
shown in Figure 11.3, and that we make the change of variables given by (11.10)
and (11.11). Show that(y 1 ,y 2 )will be distributed according to (11.12).
11.5 ( ) www Letzbe aD-dimensional random variable having a Gaussian distribu-
tion with zero mean and unit covariance matrix, and suppose that the positive definite
symmetric matrixΣhas the Cholesky decompositionΣ=LLTwhereLis a lower-
triangular matrix (i.e., one with zeros above the leading diagonal). Show that the
variabley=μ+Lzhas a Gaussian distribution with meanμand covarianceΣ.
This provides a technique for generating samples from a general multivariate Gaus-
sian using samples from a univariate Gaussian having zero mean and unit variance.
11.6 ( ) www In this exercise, we show more carefully that rejection sampling does
indeed draw samples from the desired distributionp(z). Suppose the proposal dis-
tribution isq(z)and show that the probability of a sample valuezbeing accepted is
given by ̃p(z)/kq(z)where ̃pis any unnormalized distribution that is proportional to
p(z), and the constantkis set to the smallest value that ensureskq(z) ̃p(z)for all
values ofz. Note that the probability of drawing a valuezis given by the probability
of drawing that value fromq(z)times the probability of accepting that value given
that it has been drawn. Make use of this, along with the sum and product rules of
probability, to write down the normalized form for the distribution overz, and show
that it equalsp(z).
11.7 ( ) Suppose thatzhas a uniform distribution over the interval[0,1]. Show that the
variabley=btanz+chas a Cauchy distribution given by (11.16).
11.8 ( ) Determine expressions for the coefficientskiin the envelope distribution
(11.17) for adaptive rejection sampling using the requirements of continuity and nor-
malization.
11.9 ( ) By making use of the technique discussed in Section 11.1.1 for sampling
from a single exponential distribution, devise an algorithm for sampling from the
piecewise exponential distribution defined by (11.17).
11.10 ( ) Show that the simple random walk over the integers defined by (11.34), (11.35),
and (11.36) has the property thatE[(z(τ))^2 ]=E[(z(τ−1))^2 ]+1/ 2 and hence by
induction thatE[(z(τ))^2 ]=τ/ 2.
