Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 / 1 / ii

Let

V = (X_{1} (t_{1}), X_{1} (t_{2}))^{T}

Cov (a_{1 k}, V) = λ_{k} (ϕ_{k} (t_{1}), ϕ_{k} (t_{2}))

and

Cov (V) = Σ = (c_{X} (t_{1}, t_{1}) c_{X} (t_{2}, t_{1}) c_{X} (t_{1}, t_{2}) c_{X} (t_{2}, t_{2})) .

E [a_{1 k} ∣ X_{1} (t_{1}), X_{1} (t_{2})] = λ_{k} (ϕ_{k} (t_{1}), ϕ_{k} (t_{2})) Σ^{- 1} (X_{1} (t_{1}) X_{1} (t_{2})) .

Σ

is singular, the same formula uses its Moore-Penrose pseudoinverse.

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