Changing $X_k$ while keeping all other coordinates fixed changes every candidate linear form by at most
$$
2\max_{1\leq\ell\leq m}|A_{k,\ell}|.
$$
The maximum of finitely many functions obeys the same bound, so part b applies with $c_k=2\max_\ell|A_{k,\ell}|$. Therefore
$$
\operatorname{Var}(Z)
\leq\sum_{k=1}^n
\left(\max_{1\leq\ell\leq m}|A_{k,\ell}|\right)^2.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-208/2/c/solution