 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-216/4/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

Z_{i}

and

ξ_{i}

imply that their linear combination is also multivariate normal. Its expected value is zero and its covariance matrix is

Cov (Y_{i}) = Λ Cov (Z_{i}) Λ^{T} + Cov (ξ_{i}) = Λ Λ^{T} + σ^{2} I_{p} .

Therefore

Y_{i} \sim N_{p} (0, Λ Λ^{T} + σ^{2} I_{p}), i = 1, \dots, n, independently .

The noise variance is assumed positive. The factor contribution is positive semidefinite and has rank at most

k

, while the noise makes the marginal covariance nonsingular.

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