Monte Carlo Estimation and Simulation
James Keesling
In earlier sections of the course we have presented several ways to estimate integrals.
These include,
NewtonCotes Closed
and
NewtonCotes Open Formulas
,
Gaussian Quadra
ture
, and
Romberg Integration
. In these methods we used exact algorithms and could cal
culate the error in the estimate in a precise way. The method we now introduce involves
the
random
choice of points and a
statistical
estimate of the error.
1
Using Random Numbers to Estimate Integrals
To start, we assume that we have a method for choosing points randomly from the unit
interval so that the points come from a probability density function that is constant one
on the interval. For practical purposes, the command
rand
() in your TI89 will suffice.
Suppose that we wish to estimate the integral
R
b
a
f
(
x
)
dx
.
The
Monte Carlo Method
does this by first choosing
n
random points from the unit interval [0
,
1]. Call this set of
points
{
x
i
}
n
i
=1
. Our formula for the estimate is given by the following calculation.
b

a
n
·
n
X
i
=1
f
(
x
i
)
Think of the formula as calculating the average value of the function
f
(
x
) and multi
plying this value by the length of the interval [
a, b
]. The average value for the function is
determined by a random sample of the points in the interval [
a, b
]. The term
Monte Carlo
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 Summer '08
 Kyung
 Statistics, Normal Distribution, Probability theory, probability density function, random numbers, TI, exponential waiting times

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