# Exercise Solution 4.3

- We first prove the technical result. This involves no random quantities. It is just algebra:
[s1]

[s2]

[s3]

[s4]

[s5]

- We next use our technical result to determine the bias of the sample variance estimator
[4.5]

The derivation is

[s6]

[s7]

[s8]

[s9]

[s10]

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

[s11]

[s12]

[s13]

We conclude that sample estimator [4.5] is biased.

- Finally, we determine the bias of the alternative estimator of variance
[4.27]

which we can denote

[s14]

The derivation largely parallels that of part (b):

[s15]

[s16]

[s17]

[s18]

[s19]

[s20]

[s21]

[s22]

[s23]

Estimator [4.27] is unbiased.

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