Exercise 2.10.1 Consider a regression model Y = X +e, e N(0, 2I) and suppose that

Exercise 2.10.1 Consider a regression model Y = Xβ +e, e ∼ N(0,σ 2I) and suppose that we want to predict the value of a future observation, say y0, that will be independent of Y and be distributed N(x

0β ,σ 2).

(a) Find the distribution of y0−x

0

ˆβ



MSE 1+x

0 (XX)−1x0

.

(b) Find a 95% prediction interval for y0.

(c) Let η ∈ (0,0.5]. The 100ηth percentile of the distribution of y0 is, say, γ (η) = x
0β +z(η)σ . (Note that z(η) is a negative number.) Find a (1−α)100%