Q$2.$ A ternary confusion channel with input $x$ and output $y$ has transition probability matrix:

X

Y

$Q=\left[\begin{array}{ccc}1& \frac{1}{2}& 0\\ 0& \frac{1}{2}& 1\end{array}\right]$

Transition matrix $(x|)$

$($a$)$ Assume input has the following distribution, please find the mutual information between the input and output.

$P_x=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$

$($b$)$ Please find the channel capacity and the optimal input probability distribution that achieves the capacity.