The <bounded differences property> with constants $c_i$ means
$$
|f(x)-f(x')|\leq c_i
$$
whenever $x$ and $x'$ differ only in coordinate $i$. Conditional on $X_{-i}$, the range of $Z$ is therefore at most $c_i$. The range bound on variance gives
$$
\operatorname{Var}(Z\mid X_{-i})\leq\frac{c_i^2}{4}.
$$
Substitution into the <Efron–Stein inequality> yields
$$
\operatorname{Var}(Z)\leq\frac14\sum_{i=1}^nc_i^2.
$$

Solution

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

 Solution

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