MONTE CARLO ERROR
1
Error 
n
b
or ò
a
b-a
f ( x)dx »
f ( xi )
å
n i =1
n
Let 𝑌𝑖 be the mean of 𝑏 − 𝑎 𝑓 𝑥𝑖 .
Note that 𝑌1 = 𝑌2 = ⋯ = 𝑌𝑛 = 𝜇.
𝑌1 + 𝑌2 + ⋯ +𝑌𝑛
𝑛𝜇
𝐸
=
=𝜇
𝑛
𝑛
Variance:
𝑉𝑎𝑟
𝑌1 +𝑌2 +⋯+𝑌𝑛
𝑛
Since 𝑌𝑖 ’s are independent
𝑌1
𝑌2
𝑉𝑎𝑟
+ 𝑉𝑎𝑟
+ ⋯ + 𝑉𝑎𝑟
𝑛
=
1
𝑉𝑎𝑟
2
𝑛
𝑌1 +
𝑛
1
𝑉𝑎𝑟
2
𝑛
𝑌𝑛
𝑛
𝑌2 + ⋯ +
1
𝑉𝑎𝑟
2
𝑛
𝑌𝑛 .
Note that 𝑌𝑖 ’s have equal variances, say 𝜎 2 ,
1
𝑛2
𝑛
𝑖=1 𝑉𝑎𝑟
𝑌𝑖 =
1
2
𝑛𝜎
𝑛2
=
𝜎2
.
𝑛
b
or ò
a
b-a
f ( x)dx »
f ( xi )
å
n i =1
n
Standard deviation is
𝜎
𝑛
and we can decrease it by
increasing 𝑛 at the rate proportional to
1
.
𝑛
Variance Reduction Techniques
-techniques for improving the speed or efficiency
of a Monte Carlo Simulation