Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 326 / 4 / 3

Solution

Put

c_{i} = \int_{D} φ_{i} (x) d λ_{n} (x)

Z = \int_{D} U (x) d λ_{n} (x) = \sum_{i = 1}^{k} ν_{i} c_{i} ξ_{i} .

Var (Z) = \frac{1}{2} i = 1 \sum k ν_{i} (\int_{D} φ_{i} d λ_{n})^{2} .

Finally, orthonormality of the

φ_{i}

gives

∥ U ∥_{X}^{2} = \sum_{i = 1}^{k} ν_{i} ξ_{i}^{2} .

Since

E ξ_{i}^{2} = 1/2

\int_{Ω} ∥ U ∥_{X}^{2} d P = \frac{1}{2} i = 1 \sum k ν_{i} .

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