**for**

**Background**

We start the familiar example of finding the
area of a circle. Figure 1 below shows a circle with
radius
inscribed within a square. The area of the circle
is , and
the area of the square is . The
ratio of the area of the circle to the area of the square
is

**Figure
1.**

If we could compute ratio, then we could
multiple it by four to obtain the value . One
particularly simple way to do this is to pick lattice points in the
square and count how many of them lie inside the circle, see Figure
2. Suppose for example that the points
are selected, then there are 812
points inside the circle and 212
points outside the circle and the percentage of points inside the
circle is . Then
the area of the circle is approximated with the following
calculation

**Figure
2.**

**Monte Carlo Method for
**

Monte Carlo methods can be thought of as
statistical simulation methods that utilize a sequences of random
numbers to perform the simulation. The name "Monte Carlo''
was coined by Nicholas
Constantine Metropolis (1915-1999) and inspired by
Stanslaw
Ulam (1909-1986), because of the similarity of statistical
simulation to games of chance, and because Monte Carlo is a center
for gambling and games of chance. In a typical process one
compute the number of points in a set **A**
that lies inside box **R**. The
ratio of the number of points that fall inside **A**
to the total number of points tried is equal to the ratio of the two
areas (or volume in 3 dimensions). The accuracy of the
ratio
depends on the number of points used, with more points leading to a
more accurate value.

A simple Monte Carlo simulation to
approximate the value of could
involve randomly selecting points in
the unit square and determining the ratio , where
is number of points that satisfy . In
a typical simulation of sample size
there were
points satisfying , shown
in Figure 3. Using this data, we obtain

and

**Figure
3.**

Every time a Monte Carlo simulation is made using the same sample size it will come up with a slightly different value. The values converge very slowly of the order . This property is a consequence of the Central Limit Theorem.

**Proof** **Monte
Carlo Pi**

**Remark.** Another
interesting simulation for approximating is
known as Buffon's
Needle problem. The reader can find many
references to it in books, articles and on the
internet.

**Computer
Programs** **Monte
Carlo Pi **

**Mathematica Subroutine (Monte Carlo
Pi).**

**Example 1.** Use
Monte Carlo simulation to approximate the
number .

**Solution
1.**

**Area Under a Curve**

Monte Carlo simulation can be used to
approximate the area
under a curve for . First,
we must determine the rectangular box
containing
as follows.

where .

Second, randomly pick points in
,
where
are chosen from independent uniformly distributed random variables
over ,
respectively. Third, calculate the ratio
as follows:

,
where
is number of points that lie in .

The area is computed using the approximation, , and
we can use the formula

.

An "estimate" for the accuracy of the above computation is

.

**Algorithm Monte Carlo
Simulation.** Given
that for . Use
Monte Carlo simulation to approximate the integral

.

**Mathematica Subroutine (Monte Carlo
Simulation).**

**Caveat.** The above
subroutine is for pedagogical purposes. The standard
implementation of Monte Carlo integration is done by selecting points
only in the domain of the function, for a one dimensional integral we
would use random numbers to select points
in the interval
and then use the approximation

.

This idea will be developed in the module Monte
Carlo integration.

**Example 2.** Use
Monte Carlo simulation to approximate the
integral

**Solution
2.**

**Example 3.** Use
Monte Carlo simulation to approximate the
integral

**Solution
3.**

**Example 4.** Use
Monte Carlo simulation to approximate the
integral

**Solution
4.**

**Question.** Which of
the above three examples will give better results if the goal is to
obtain a numerical approximation of

**Answer.**

**Example 5.** Use
Monte Carlo simulation to approximate the
integral .

**Solution
5.**

**Area of a region defined by
inequalities.**

Monte Carlo simulation can be used to
approximate the area of a region defined by a set of inequalities or
constraints.

**Mathematica Subroutine (Monte Carlo
Simulation).**

**Example 6.** Use
Monte Carlo simulation to approximate the area of the cardioid
defined by the constraint

.

**Example 7.** Use
Monte Carlo simulation to approximate the area of the region defined
by the constraints

**Research Experience for
Undergraduates**

**Monte
Carlo Pi** Internet hyperlinks to web sites
and a bibliography of articles.

**Download this
Mathematica Notebook****
****Monte
****Carlo
Pi**

(c) John H. Mathews 2005