Past exam of the communication Theory course of the University of Oxford / 2010 / 2 / b / i / Solution

Choose a most probable symbol $x_{\ast }$, and let $B=1_{\{X=x_{\ast }\}}$. Its binary entropy isSince $B$ is a function of $X$, part (a)(ii) gives $H(X)\ge h_2(\Pi )$. On $[1/2,1]$, the concave function $h_2$ lies above the chord joining $(1/2,1)$ and $(1,0)$. That chord has height $2(1-\Pi )$. ThereforeThe endpoint $\Pi =1$ is included by continuity and corresponds to a deterministic source.