Past exam of the communication Theory course of the University of Oxford / 2007 / 1 / c / Solution

Independence gives $P(X=Y)={\sum }_r{p}_r{q}_r={\mathbb{E}}_p[2^{{\log }_2{q}_X}]$. The given exponential convexity inequality therefore yieldsTaking complementary probabilities provesThe strict positivity of the entries of $q$ makes all logarithms appearing in this finite expectation well defined.