Suppose X 1,…,Xnnare iid random variables with mean μ and variance σ 2 but with unknown distribution. Show that the sample variance S 2
= n−1
∑ i=1
n
​
(X i
​
− X
ˉ
) 2
​
= n−1
∑ i=1
n
​
X i
2
​
−n X
ˉ
2
​
is an unbiased estimator for σ 2
, i.e. show, using results on the linearity of expectation (e.g. E(a+bX)=a+bE(X)), that E(S 2
)=σ 2
. NB: when the X i
​
are not normal, (n−1)S 2
/σ 2
is not χ n−1
2
​
.