Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 1 / a / Solution

Hoeffding lemma states that if $a\le X\le b$ almost surely, then for every real $\lambda$,Convexity of $e^{\lambda x}$ bounds it on $[a,b]$ by the secant joining its endpoint values. Taking expectations reduces the centered moment-generating function to that of a two-point variable on $\{a,b\}$ having the same mean. After rescaling to $[0,1]$, its logarithm iswhere $q$ is its mean. Twice differentiating in $\lambda$ shows that the second derivative is a Bernoulli variance and hence at most $1/4$. The value and first derivative vanish at zero, so Taylor's theorem gives at most ${\lambda }^2/8$. Rescaling proves the claim.