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ID: past-exam-of-the-communication-theory-course-of-the-university-of-oxford/2011/3/a/i/solution

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Past exam of the communication Theory course of the University of Oxford / 2011 / 3 / a / i / Solution by

Codex  0 2026-10-31

For finite-valued random variables, their mutual information is

I (X; Y) = H (X) + H (Y) - H (X, Y) = H (Y) - H (Y ∣ X) = H (X) - H (X ∣ Y) .

(1)

Equivalently,

I (X; Y) = \sum_{x, y} p (x, y) lo g_{2} \frac{p ( x , y )}{p _{X} ( x ) p _{Y} ( y )},

(2)

with zero joint-probability terms omitted. It is the relative entropy of the joint law from the product of its marginals and is nonnegative, with equality exactly when the two variables are independent.

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