We would like to evaluate the following definite integral,
where we assume that 
is real-valued on 
.
Figure 9
depicts a typical integral to be evaluated.
The idea is to manipulate the definite integral into a form that can be solved
by Monte Carlo. To do this, we define the following function on 
,
and insert into Eq. (64) to obtain the following expression for
the integral 
:
Note that 
can be viewed as a uniform pdf on the interval , as
depicted in
Figure 9.
Given that is a pdf, we observe that the integral on the right
hand side of Eq. (66) is simply the expectation value for :
We now draw samples 
from the pdf , and for each we will
evaluate 
and form the average 
,
But Eq. (62) states the expectation value for the average of 
samples is the
expectation value for , 
, hence
Thus we can estimate the true value of the integral on by
taking the
average of observations of the integrand, with the r.v. 
sampled
uniformly over the interval .
For now, this implies that the interval is finite,
since an infinite interval cannot have a uniform pdf.
We will see later that infinite ranges of integration can be accommodated with
more sophisticated techniques.
Recall that Eq. (63) related the true variance in the average to the true
variance in 
,
Although we do not know
,
since it is a property of the pdf
and the
real function , it is a constant.
Furthermore, if we associate the error in
our estimate of the integral with the standard deviation, then we might
expect the error in the estimate of to decrease by the factor 
.
This will be shown more rigorously later when we consider the Central Limit
Theorem, but now we are arguing on the basis of the functional form of
and a hazy correspondence of standard deviation with ``error''.
What we are missing is a way to estimate
,
as we were able to
estimate 
with .
