Define the function
h(x) = sin(2711 2.00 z:)| (cos(2732 L300 =3)) ?
for x = (21,...,1100) R100 (z is 100-dimensional.) (Python code for evaluating h will be posted on
Blackboard.]
(a) Compute an approximate 95% confidence interval for the value of
h(x)dx
10.1100
The interval should be no wider than 0.01. (Recall that (0, 1100 is the 100-dimensional unit hypercube.)
(b) Compute an approximate 95% confidence interval for the value of
h(x)da
310,1.05100
The interval should be no wider than 1. Your answer for part (b) should be a lot larger than for part (a).
Why?
(c) Compute an approximate 95% confidence interval for the value of
h(x) da
110,1100B
where B is the ball of radius 6 in R100. The interval should be no wider than 0.01.
(d) What goes wrong if you try to do the previous problem using the ball of radius 4?
(e) (Extra credit) What is it about the function h and about the domain of integration that allows Monte
Carlo to work in 100 dimensions in parts (a)-(e) for only moderate amounts of computation?
(f) (Hard extra credit) Compute an approximate 95% confidence interval for the value of
I h(a)da
10.1100B
where B is the ball of radius 4 in R100. The interval should be no wider than 10-11 and should not
contain 0. You will need to use importance sampling and to choose the proposal distribution cleverly.