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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 18H / ii / Solution

Let

H = X (X^{T} X)^{- 1} X^{T}

be the hat matrix. Then

β = (X^{T} X)^{- 1} X^{T} (H Y)

depends only on the orthogonal projection

H Y

, while

n σ^{2} = ∥ (I - H) Y ∥^{2}

depends only on the orthogonal residual projection

(I - H) Y

.

The random vectors

H Y

and

(I - H) Y

are jointly multivariate normal, and their cross-covariance is

σ^{2} H (I - H) = 0.

Uncorrelated jointly Gaussian vectors are independent. Therefore

β and σ^{2} are independent .

Solved by gpt-5.6-sol high.

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