Ex:           Three random variables, X1, X2, and X3, are independent and uniformly distributed on [−1/512, 1/512]. (They represent the distributions of errors for an 8-bit analog-to-digital converter that quantizes voltages by rounding off to the nearest 1/28 V.) Find the mean and variance of Z = (X1 + X2 + X3)/3. (This represents the result of oversampling by a factor of three and averaging, as compared to using a single sample.)

Sol'n:      We use the following tools for linear combinations of random variables (extended from formulas for the sum of two random variables):

and, (for independent random variables)

First, we observe that the means of X1, X2, and X3 are zero since they are uniform distributions centered at x = 0. Thus, we can calculate the mean of Z as follows:

Second, we observe that one may obtain each of the Xi from a uniform distribution on (0,1) that is shifted by −1/2, (which doesn't change the variance), and then scaling by a = 1/256. Since the variance for a uniform distribution on (0,1) is 1/12, we have the following variance for each of the Xi:

Now we apply the formula for the variance of a sum of independent random variables to Z where a = b = c = 1/3:

or

or

Note that we have a factor of 1/9 for each term but we have 3 terms. Thus, our final answer contains only a factor of 3 in the denominator.