Independence makes <varentropy> additive. For a Bernoulli variable,
$$
V_H(X_i)=p_i(1-p_i)
\left(\log\frac{p_i}{1-p_i}\right)^2.
$$
Part (c) bounds the logarithm by $|\log\delta_i|$, while $p_i(1-p_i)\leq1/4$. Therefore
$$
V_H(X_1,\ldots,X_n)=\sum_iV_H(X_i)
\leq\frac14\sum_{i=1}^n(\log\delta_i)^2.
$$

Solution

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

{title2=$V_H(X)$}
{wiki}

The varentropy of a discrete random variable with mass function $P$ is the variance of its information content:
$$
V_H(X)=\operatorname{Var}[-\log P(X)].
$$


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