 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/o/o2/b/ii/solution

c

\sum_{i} z_{i} = 0

. Put

S = \sum_{i} z_{i}^{2} > 0

. Then

X^{T} X = (n 0 0 S) .

Part (a) therefore gives

Cov (α β) = σ^{2} (1/ n 0 0 1/ S) .

In particular the two estimates are uncorrelated, and under the assumed normal-error model they are independent.

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