Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 2 / c

Table of contents
- Solution c

Solution

Changing

X_{k}

while keeping all other coordinates fixed changes every candidate linear form by at most

2 max_{1 \leq ℓ \leq m} ∣ A_{k, ℓ} ∣.

The maximum of finitely many functions obeys the same bound, so part b applies with

c_{k} = 2 max_{ℓ} ∣ A_{k, ℓ} ∣

. Therefore

Var (Z) \leq \sum_{k = 1}^{n} (max_{1 \leq ℓ \leq m} ∣ A_{k, ℓ} ∣)^{2} .

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