Past exam of the communication Theory course of the University of Oxford / 2009 / 3 / a / Solution

For finite-valued variables, mutual information isTerms of zero joint probability are zero. A discrete memoryless channel has fixed transition probabilities $W(y\mid x)$ on finite alphabets, and conditional on an input sequence its outputs are independent with the product law ${\prod }_{i}W(y_i\mid x_i)$. Its channel capacity isbits per channel use. The maximum is over all input distributions, with the transition law held fixed.