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**.