= Capacity after independent output erasure
{title2=$C_{\mathrm{erased}}=(1-p)C$}
Pass the output of a <discrete memoryless channel> of <channel capacity> $C$ through an independent <finite-alphabet erasure channel> with erasure probability $p$. Its capacity becomes $(1-p)C$.
To prove it, let $X$ be the original input, $Y$ the original output, $E$ the erasure indicator, and $Z$ the final output. Since the erasure marker is distinguishable, $E$ is determined by $Z$. It is independent of $X,Y$, so the <chain rule for mutual information> gives $I(X;Z)=I(X;E,Z)=I(X;Z\mid E)=(1-p)I(X;Y)$. Maximization over the same input distributions proves the formula. For $p<1$ the maximizing distributions are exactly the original channel's maximizers. For $p=1$, every input distribution attains the zero capacity. Input-dependent erasure need not satisfy this scaling identity.
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