Now let us draw 
samples 
from the pdf 
and
define the
following linear combination of the samples:
where the parameters 
are real constants and the 
are
real-valued functions.
Since the 
are r.v.'s, and 
is a linear combination of functions of the
r.v.'s, is also a r.v. We now examine the properties of , in particular
its expectation value and variance. Referring to our earlier discussion of the
mean and variance of a linear combination, expressed as Eq. (9) and
Eq. (15),
respectively, we find
Now consider the special case where 
and 
:
Note that is simply the average value of the sampled r.v.'s.
Now consider
the expectation value for , using Eq. (61):
In other words, the expectation value for the average (not
the average itself!)
of observations of the r.v. 
is simply the expectation value for
.
This statement is not as trivial as it may seem, because we may not know
in
general, because is a property of and the pdf .
However,
Eq. (62) assures us that an average of observations of
will be a
reasonable estimate of .
Later, we will introduce the concept of an unbiased estimator, and suffice to
say for now, that Eq. (62) proves that the simple average is an unbiased
estimator for the mean.
Now let us consider the variance in , in particular its dependence on the
sample size.
Considering again the case where and , and
using
Eq. (60), the variance in the linear combination is given by:
Hence the variance in the average value of samples of is a factor
of
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 samples of , denoted ,
Eq. (62)
tells us
that the true mean for is equal to the true mean for .
On the other hand,
Eq. (63) tells us that the true variance for 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.
