Let X = (X1, ..., Xn) be a random sample from a normal N(μ, σ^2) distribution, where the mean μ and the variance σ^2 are unknown.
(a) Show that Xn̅ = n^-1 ∑ Xi is a consistent estimator of μ.
(b) Show that Sn^2 = n^-1 ∑(Xi - Xn̅)^2 is a biased estimator of σ^2 with the bias b(σ^2) = -σ^2/n. (Hint: You may assume, without loss of generality, that μ = 0.)
(c) Are Xn̅ and Sn^2 independent? Why or why not?