Let $P_i$ be the marginal <probability distribution> of $X_i$. <Subadditivity of information entropy> and the <concavity of information entropy> give
$$
H(X_1^n)\leq\sum_{i=1}^nH(P_i)
\leq nH\!\left(\frac1n\sum_{i=1}^nP_i\right).
$$
For each $a\in A$,
$$
\frac1n\sum_{i=1}^nP_i(a)
=\mathbb P(X_J=a)=\overline P(a),
$$
so $H(X_1^n)\leq nH(\overline P)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 224 / 2 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-224/2/d/solution