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.