 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/27k/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

A

, invariance in distribution gives

φ_{X} (A^{T} u) = φ_{A X} (u) = φ_{X} (u) .

The orthogonal group is transitive on each Euclidean sphere, so

φ_{X} (u)

depends only on

∥ u ∥^{2}

f (r) = φ_{X} (r, e_{1}), r \geq 0.

f

continuous. Taking

A = - I

also gives

X = d - X

, and hence

φ_{X} (u) = φ_{X} (- u) = \overline{φ_{X} (u)},

f

is real-valued. Thus

φ_{X} (u) = f (u_{1}^{2} + \dots + u_{d}^{2}),

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