SA421 – Simulation Modeling
Asst. Prof. David Phillips
Fall 2013
Homework on Antithetical Variables
1. Consider the problem of estimating
Z
ln(2) =
0
1
1
dx
1+x
using simulation. Conduct a variance
analysis of using antithetical variables similar to the
R1
classroom analysis of estimating 0 ex dx. To do this, perform the following steps.
(a) Calculate the variance when taking two samples of a naive simulation, i.e., calculate
1
1
Var (
+
)/2
1 + X1 1 + X2
where X1 , X2 are independent and uniform random variables on the interval [0, 1].
(b) Then, calculate the variance for one iteration of an antithetical approach, i.e., calculate
1
1
1
1
+
)/2 = Var (
+
)/2
Var (
1 + X1 1 + (1 − X1 )
1 + X1 2 − X1
where X1 is a uniform random variable on the interval [0, 1].
2. Let Xi , i = 1, . . . , 5 be independent exponential random variables with mean 1 and consider
" 5
#
X
θ=P
iXi ≥ 21.6 .
i=1
(a) Explain how to use simulation to estimate θ.
(b) Explain how to use antithetic variables in your simulation.
(c) Use Excel to determine if this is results in variance reduction or not.
1