Exercise Solution 4.3

  1. We first prove the technical result. This involves no random quantities. It is just algebra:






  2. We next use our technical result to determine the bias of the sample variance estimator


    The derivation is






    To obtain the next step in the derivation, we apply the result of Exercise 3.15 twice:




    We conclude that sample estimator [4.5] is biased.

  3. Finally, we determine the bias of the alternative estimator of variance


    which we can denote


    The derivation largely parallels that of part (b):










    Estimator [4.27] is unbiased.