= Solution
Every row has <Shannon entropy> $1$, so $H(Y\mid X)=1$ for every input law. Since the output alphabet has three symbols, <Gibbs inequality> gives $H(Y)\leq\log_23$. The uniform input gives
$$
q=\left(\frac{p_1+p_2}2,\frac{p_1+p_3}2,\frac{p_2+p_3}2\right)=(1/3,1/3,1/3),
$$
so it attains the upper bound. Therefore
$$
\boxed{p_X=(1/3,1/3,1/3),\qquad C=\log_23-1=\log_2(3/2).}
$$
Uniform output is necessary for equality in the <Shannon entropy>[entropy] bound, and the three displayed equations then force the uniform input, so this optimum input law is unique.
Back to article page
