We now draw samples 
from the pdf 
, and for each 
we will
evaluate 
and form the average G,
But Eq. (62) states the expectation value for the average of N samples is the
expectation value for 
, 
, hence
Thus we can estimate the true value of the integral I on 
by
taking the
average of N observations of the integrand, with the r.v. x 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 G to the true variance in g,
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 I with the standard deviation, then we might
expect the error in the estimate of I 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 G.
