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2.2.4 Variance of Linear Combination

The mean of a linear combination of r.v.'s is the linear combination of the means, as shown in Eq. (9), because the mean is a linear statistic, as is clear from Eq. (7). On the other hand, the variance is clearly not a linear statistic, since the r.v. is squared. However, we will find it necessary to consider the variance of a linear combination of r.v.'s, and it is straightforward to show the following:

Let us consider the average value of the product of two r.v.'s:

Now if x and y are independent r.v.'s, then

where is the probability for the r.v. to occur. But if Eq. (17) is inserted into Eq. (16), we find

Thus, if two r.v.'s are independent, the expectation value of their product is the product of their expectation values. Now consider the case of the variance of a linear combination of r.v.'s given in Eq. (15), and note that if the r.v.'s g and h are independent, Eq. (18) when inserted into Eq. (15) yields the following expression, valid only when and are independent r.v.'s: