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2.4.6 Sums of Random Variables     continued...

Now consider the special case where and :

Note that G is simply the average value of the N sampled r.v.'s. Now consider the expectation value for G, using Eq. (61):

In other words, the expectation value for the average ( not the average itself!) of N 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 N 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 G, in particular its dependence on the sample size.