For $X\sim\operatorname{Bernoulli}(p)$,
$$
H(X)=p\log\frac1p+(1-p)\log\frac1{1-p}.
$$
The inequality $-\log(1-p)\leq p/(1-p)$ bounds the second term by $p$. Also $\log x\leq2\sqrt x$ for $x\geq1$, so $p\log(1/p)\leq2\sqrt p$. Hence
$$
H(X)\leq2\sqrt p+p.
$$


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-208/1/b/solution