Tutorial on Monte Carlo Techniques
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2019-02-22
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Tutorial on Monte Carlo Techniques
Gabriel A. Terejanu
Department of Computer Science and Engineering
University at Buffalo, Buffalo, NY 14260
terejanu@buffalo.edu
1 Introduction
Monte Ca rlo (MC) technique is a numerical method that makes use of random numbers to s olve
mathematical problems for which an analytical solution is not known. The first article, “Th e Monte
Carlo Method” by Metropolis and Ulam, has appeared for the first time in 1949 [9], even though well
before that certain statistical problems were solved using random numbers. Since the simulation of
random numbers is very time consum ing, MC has became practical only w ith the advent of computers.
A simple MC simulation is the determination of π. Suppose we have a circle with radius r = 1
inscribed within a square. Then the area ratio is:
A
circle
A
square
=
πr
2
4r
2
=
π
4
(1)
To determ ine π we randomly select n points, p
i
= (x
i
, y
i
) for i = 1 . . . n, in the square and approximate
the area ratio by:
A
circle
A
square
=
m
n
(2)
where m is the number of random points in the circle, they must satisfy x
2
i
+y
2
i
≤ 1. Hence π = 4×
m
n
.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 1: Compute π
This method is called hit-or-miss Monte Carlo since the estimate is computed as the actual ratio
of hits to random tries. It is th e least efficient MC method.
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