Description
1. Exercises 5.12, and 5.14.
2. Given two random variables X and Y , prove the law of total variance
var(Y ) = E{var(Y |X)} + var{E(Y |X)}.
Be explicit at every step of your proof.
3. Define θ =
R
A
g(x) dx, where A is a bounded set and g ∈ L2(A). Let f be an
importance function which is a density function supported on the set A.
(a) Describe the steps to obtain the importance sampling estimator ˆθn, where n is
the number of random samples generated during the process.
(b) Show that the Monte Carlo variance of ˆθn is
var(ˆθn)x =
1
n
Z
A
g
2
(x)
f(x)
dx − θ
2
(c) Show that the optimal importance function f
∗
, i.e., the minimizer of var(ˆθn), is
f
∗
(x) = |g(x)|
R
A
|g(x)| dx
,
and derive the theoretical lower bound of var(ˆθn).
1
