Suppose there are three independent observations that are to be averaged. The known variances of the three observations are 4, 9, and 16. Two different averages are proposed, the simple arithmetic...

Suppose there are three independent observations that are to be averaged. The known variances of the three observations are 4, 9, and 16. Two different averages are proposed, the simple arithmetic average and the weighted average where each observation is weighted by the reciprocal of its variance. Use variances of linear functions, equation 3.21 and following, to demonstrate that the weighted average has the smaller variance. Can you find any other weighting that will give an even smaller variance?