Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 208 / 2

Put $\mu =\mathbb{E}Z$ and $h(u)=(1+u)\log (1+u)-u$. For $\lambda >0$, the Chernoff bound and the assumed cumulant-generating-function estimate giveThe optimizer satisfies $e^{\lambda }=1+t/\mu$, and hence

For the left tail, apply the same argument at a negative parameter. If $0<t<\mu$ the optimizer satisfies $e^{-\lambda }=1-t/\mu$, givingFor $t\ge \mu$, nonnegativity of $Z$ makes the strict lower-tail event empty, with the boundary handled directly.

Solved by gpt-5.6-sol high.

Let $d_i=f(X)-f_i(X^{(i)})$. The self-bounding function assumptions say $0\le d_i\le 1$ and ${\sum }_i{d}_i\le Z$. Apply tensorization of entropy to $e^{\lambda Z}$ and the one-coordinate entropy inequality. Since $\phi (u)=e^u-u-1$ is convex and $\phi (td)\le d\phi (t)$ for $0\le d\le 1$, the resulting bound is

Writing $\psi (\lambda )=\log \mathbb{E}{e}^{\lambda (Z-\mu )}$ and dividing by the moment-generating function reduces this toBoth sides have finite limits at zero and $\psi (0)={\psi }^{\prime }(0)=0$. Integrating from zero to $\lambda$, with the direction interpreted correctly when $\lambda <0$, yieldswhich is the required inequality.

Solved by gpt-5.6-sol high.