= Solution
<Shannon second coding theorem> identifies the <channel capacity> of a discrete memoryless channel as $C=\max_{P_X}I(X;Y)$: rates below $C$ admit block codes with error probability tending to zero, whereas a rate above $C$ cannot achieve vanishing error. Entropies here are measured in bits. For a <binary erasure channel>, a non-erased output determines its input exactly, while an erasure, independent of the input, leaves the original uncertainty unchanged. If $\mathbb P(X=1)=q$, then
$$
H(X)=h_2(q),\qquad H(X\mid Y)=p h_2(q),\qquad
I(X;Y)=(1-p)h_2(q).
$$
The <binary entropy function> has maximum one at $q=1/2$. Therefore
$$
\boxed{C=1-p\ \text{bits per channel use}}.
$$
This includes the noiseless case $p=0$ and the wholly erased case $p=1$.
Back to article page
