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HW3 (Due 10/18)
For \(Y\) and \(X\) take the value of 1,2, and 3, please find the capacity when the transition matrix \(T\) (\(p_{Y|X}(i|j)=T_{i,j}\)) is given by
\(T = \begin{pmatrix} 1/3 & 1/3 & 1/3 \\ 1/3 & 1/3 & 1/3 \\
1/3 & 1/3 & 1/3 \end{pmatrix}\)
\(T = \begin{pmatrix} 1/2 & 1/2 & 0 \\ 0 & 1/2 & 1/2 \\
1/2 & 0 & 1/2 \end{pmatrix}\)
Another question on channel capacity, given a channel with input alphabets \(\{1,2,\cdots,n\}\) and output alphabets \(\{1,2,\cdots,m\}\). If the probability of output \(Y\) given input \(X\) is always equal to some permutation of \(p_1, p_2,\cdots,p_m\), i.e., \(p_{Y|X}(i|j) = p_{\sigma_j(i)}\) for some permutation \(\sigma_j(\cdot)\).
For example, \(p_1 = 0.2, p_2 = 0.3, p_3=0.5\) and \(\sigma_1(1)=2, \sigma_1(2)=1, \sigma_1(3)=3\). So \(p_{Y|X}(1|1)=p_{\sigma_1(1)}=p_2=0.3\), \(p_{Y|X}(2|1)=p_{\sigma_1(2)}=p_1=0.2\), and \(p_{Y|X}(3|1)=p_{\sigma_1(3)}=p_3=0.5\).
Moreover, assume that \(\sum_{j=1}^{m} p_{Y|X}(i|j) =
\sum_{j=1}^{m} p_{Y|X}(i’|j)\) for any \(i\) and \(i'\).
Show that the capacity of the channel is
\( \log m - H(p_1,p_2,\cdots,p_m),\)
where \(H(p_1,p_2,\cdots,p_m) = - \sum_{i} p_i \log p_i \).
If the average weight of the 10 packages is NOT more than 1 Kg, and without any other information is given,
What is the maximum possible weight of the lightest package?
What is the maximum possible weight of the heaviest package?
What is the maximum possible weight of the 5th heaviest package?
What is the maximum possible weight of the 3rd heaviest package?
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