Past exam of the communication Theory course of the University of Oxford / 2008 / 1 / b / Solution

The conditional output distributions are permutations of the same triple, so both conditional entropies equalWith input probability $p$, the three output probabilities are $q_1=(2-p)/6$, $q_2=(1+2p)/6$, and $q_3=(3-p)/6$. ThereforeDifferentiating, the constant terms cancel because $\sum q_j^{\prime }=0$:Thus there is a unique maximum. Its stationary equation reduces to $3p^2+9p-5=0$, whose root in $[0,1]$ isThe other root is negative; strict concavity proves this is the global maximum.