The previous talk in the series is here. The next talk in the series is here.
In this video I present the formula for the entropy of a random variable that takes values in a finite set, prove that it satisfies the entropy axioms, and prove that it is the only formula that satisfies the entropy axioms. We first give the formula for entropy and prove that it satisfies the axioms. Next, we show that non-negativity of entropy follows from the axioms, then compute the entropy of a random variable that is uniform on a set of size n. Finally, we show that the formula for entropy follows from the axioms.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.