Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 224 / 3 / b / Solution

For the direct half of the code-distribution correspondence, let $L$ be the length function of a binary prefix code and putThen $R$ is a probability mass function andThus every prefix code determines a distribution whose ideal description lengths do not exceed the codeword lengths.

Conversely, given a probability mass function $R$, setfor $R(x)>0$. Then $2^{-L(x)}\le R(x)$, so the Kraft inequality is satisfied. Its converse supplies a binary prefix code with these lengths, andIf real lengths are allowed, the ideal choice $L(x)=-{\log }_{2}R(x)$ satisfies Kraft with equality.