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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ia/paper-2/11f/a/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 2 / 11F / a / ii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

For every

λ \in R^{2}

, the scalar

λ^{T} X

is normal with mean

λ^{T} μ

and variance

λ^{T} Vλ

. The moment-generating function of a normal distribution therefore gives

M_{X} (λ) = exp (λ^{T} μ + \frac{1}{2} λ^{T} Vλ) .

(1)

Thus $μ$ and $V$ determine the moment-generating function on all of $R^{2}$ . By the uniqueness theorem for moment-generating functions, they uniquely determine the probability distribution of $X$ .

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