Observations (xi, Yi), i = 1,..., n, are made according to the model Yi = + xi + i, where x1,...,xn are fixed constants

Observations (xi, Yi), i = 1,..., n, are made according to the model
Yi = α + βxi + εi,
where x1,...,xn are fixed constants and ε1,...,εn are iid n(0, σ2). The model is then reparameterized as
Yi = α′ + β′(xi - ) + εi.
Let  and  denote the MLEs of α and β, respectively, and ′ and ′ denote the MLEs of α′ and β′, respectively.
(a) Show that ′ = .
(b) Show that ′ ≠ . In fact, show that ′ = . Find the distribution of ′.
(c) Show that ′ and ′ are uncorrelated and, hence, independent under normality.