Given that u N(0, 2In), we showed that (n k)s2/2 2 nk. Use this fact to prove that, (a) s2 is unbiased

Given that u ∼ N(0, σ2In), we showed that (n − k)s2/σ2 ∼ χ2 n−k. Use this fact to prove that,

(a) s2 is unbiased for σ2.

(b) var(s2) = 2σ4/(n − k). Hint: E(χ2r

) = r and var(χ2r

) = 2r.