Considering again the case where 
and 
, and
using
Eq. (60), the variance in the linear combination G is given by:
Hence the variance in the average value of N samples of 
is a factor
of N
smaller than the variance in the original r.v. 
.
Note that we have yet to say
anything about how to estimate
,
only that its value decreases
as 
.
This point deserves further elaboration. The quantities 
and
are properties of the pdf 
and the real function 
.
As mentioned earlier,
they are known as the true mean and true variance, respectively,
because
they are known a priori, given the pdf 
and the function 
.
Then if we
consider a simple average of N samples of 
, denoted G,
Eq. (62)
tells us
that the true mean for G is equal to the true mean for 
.
On the other hand,
Eq. (63) tells us that the true variance for G is 
smaller than
the true variance for 
, an important consequence for estimating errors.
Later we will show how to estimate
,
an important task since in
general we don't know the true mean and variance, and these terms will have to
be estimated. Let us now apply this discussion to an important application of
Monte Carlo methods, the evaluation of definite integrals.
