My question is about the Z channel. The Z-channel has binary input and output alphabet and transitions probabilities $p(y|x)$ givin by the following matrix:
$$ Q =\pmatrix{ 1 &0\\ 1/2 &1/2}$$ we want to find the capacity of Z-channel.
Here is the solution:
I have problem in realizing the solution. For example, I'm confused about the method that $H(Y|X)$ and $H(Y)$ is computed. Why we have considered only $Pr(Y=1)$ in computing $H(Y)$? Isn't it computed using this formula:
$p(y)=\Sigma p(y|x)p(x)$ and $H(y)= p(y) \log 1/p(y) $ ?
In the equation for $H(Y)$, you mixed the standard entropy function $H$ and the highlighted binary entropy function $\textbf H$, which stands for the binary entropy function:$$\textbf H(p) = -p\log p-(1-p)\log(1-p)$$ On the other hand, the equation for $H(Y|X)$ follows directly from $$H(Y|X) = p(X=0)H(Y|X=0)+p(X=1)H(Y|X=1)$$ and noticing that $H(Y|X=0)=0$ and $H(Y|X=1)=1$.