## Turning probability into polynomials (Mark Huber, CMC)

### November 6, 2018 @ 12:15 pm - 1:10 pm

Moment generating functions (Laplace transforms) are a means for transforming probability problems into problems involving polynomials. Here I will concentrate on the binomial distribution, and use the mgf to link this distributions probabilities directly to the binomial theorem. The mgf is also a key ingredient in Chernoff bounds, which give upper bounds on the tail probabilities of binomial distributions (aka partial sums of the binomial theorem). By employing the method of smoothing and tilting, it is possible to attain bounds on the tails that go down faster than the traditional approximation heuristic that uses the Central Limit Theorem.