Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 3I / b / i

Table of contents
- Solution i

Solution

With rows indexed by outputs

A, B, ⋆

and columns by inputs

A, B

, the channel matrix is

1/2 0 1/2 0 1/2 1/2 .

This is a binary erasure channel with erasure probability

1/2

. Conditional on no erasure, the input is recovered exactly, whereas an erasure supplies no information. Thus

I (X; Y) = \frac{1}{2} H (X),

which is maximized by the uniform input distribution. Therefore

C = \frac{1}{2} bit .

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